Postgres 12

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CREATE TABLE t (geom geometry(point,26986));

INSERT INTO t 
SELECT (ST_Dump(ST_GeneratePoints('SRID=26986;POLYGON((4216635.436744224 3906228.6933973604,4225112.992686871 3914713.9388259286,4237223.322966506 3902640.089526773,4228720.260658542 3894129.3145975843,4216635.436744224 3906228.6933973604))',100,42))).geom;

CREATE INDEX idx_geom_t ON t USING gist (geom);

CREATE TABLE

INSERT 0 100

CREATE INDEX

SELECT count(*),ST_AsText(ST_Union(geom))
FROM (
  SELECT geom, ST_ClusterDBSCAN(geom,1000,2) OVER () AS cluster 
  FROM t) j
WHERE cluster IS NOT NULL
GROUP BY cluster
ORDER BY 1 DESC
LIMIT 1
-- FETCH FIRST ROW WITH TIES -- <= not supported in this fiddle (postgres 12). see above

count
st_astext

6
MULTIPOINT(4223502.405941216 3899993.0118816346,4223540.53519072 3900824.4558236054,4223549.151749019 3901611.1700444585,4224076.304240495 3900531.7731772666,4224140.724038144 3899569.014816247,4224192.121671825 3901720.1362135755)

SELECT 1

WITH i AS (
  SELECT count(*),ST_MinimumBoundingCircle(ST_Union(geom)) AS geom
  FROM (
    SELECT geom, ST_ClusterDBSCAN(geom,1000,2) OVER () AS cluster 
    FROM t) j
  WHERE cluster IS NOT NULL
  GROUP BY cluster
  ORDER BY 1 DESC
  LIMIT 1
  --FETCH FIRST ROW WITH TIES
)
SELECT geom FROM i
UNION 
SELECT geom FROM t

geom

01010000206A690000C847B59F29235041181E71D2FABA4D41

01010000206A690000A0FB64816418504100684044DED14D41

01010000206A690000B7E0A3860925504142E5240705C04D41

01010000206A69000028E7186F3D1B50417CE36E86F0C84D41

01010000206A6900003AC22C10B6205041F321BD69F5BF4D41

01010000206A690000916B8A8B82275041C66E4AB6C8C14D41

01010000206A690000CF82D0652817504158604A0B06CF4D41

01010000206A6900002A95FC64141D5041986B7E3592C84D41

01010000206A690000081AF0ADED2050411583110BABCA4D41

01010000206A690000FF2722D9BE1E504190EEC65580CF4D41

01010000206A690000387CE8647622504104C57E8CD1BF4D41

01010000206A6900008441B6498F1C50414E04C49555C44D41

01010000206A690000CB492AB6662150419DF5BAC59FCA4D41

01010000206A6900006C96E0CB7C1E504115B4FD3EA1CA4D41

01010000206A69000046EB4063CF1C5041BEB09230D0D44D41

01010000206A69000078B3D15BFE185041D76024FC19CB4D41

01010000206A690000F5AAB1747320504150194312EEC54D41

01010000206A69000060C286FFE721504175793D2D86CE4D41

01010000206A6900001EFA099ADA1E50414B3ED44146D04D41

01010000206A690000E603081C9F1C50411D5C4AABCECF4D41

01010000206A690000588255A6B41A50412F18C6C114CB4D41

01010000206A690000E02194F338195041503F2B0D7FCD4D41

01010000206A69000082C1C5E72420504102E779CD4ED74D41

01010000206A690000DE593ADE7B1C5041A383036D87D54D41

01010000206A690000310F49926D215041F03DB8DE91D44D41

01010000206A6900009AE51E06CC2550413B4DC8FD0CC74D41

01010000206A6900005B27C699E72350410B5A249CCEC84D41

01010000206A6900000F3C5F14A8205041AE3DC0BDE1C34D41

01010000206A69000052E12969181E5041736148B792D24D41

01010000206A69000089402D66D71E5041DC8F82EF68C94D41

01010000206A69000015A4562E231D5041B07FE58158C04D41

01010000206A6900001A1045416F225041BF7E3668CBC74D41

01010000206A690000C3AC6DB1D61750414F4D122E1ECF4D41

01010000206A6900008BA62B7C062950418DD5092F66C64D41

01010000206A6900002F32A84D831750410EC98C662ACC4D41

01010000206A69000002C2D83A85235041225BE5C47AC04D41

01010000206A6900009F78C907301D50414A726F118CC44D41

01010000206A690000E6DEE4BF16235041CF4B9B8487CF4D41

01010000206A6900003D783E984E1B50415C927058DDC54D41

01010000206A690000BF33AE161E1E50419CF3E4E674BD4D41

01010000206A6900001DD071D15D1B50412925652EB1D14D41

01010000206A690000BBDF77A7BA1D5041AE2811E268D14D41

01010000206A690000E1139FAC94185041F2C123AB4CC94D41

01010000206A69000037CF956E862750412D97F241F7C84D41

01010000206A690000C49C0A0579205041C7DFACB51DBD4D41

01010000206A690000B5483EF790215041AB05075209BE4D41

01010000206A69000026C031903E205041333C465D69BA4D41

01010000206A690000A9AC6CA2E523504103E4E09F5CC44D41

01010000206A690000C71E127AFB1D504132F8930CEBDC4D41

01010000206A6900006CB04ABB7219504107A5AD2420C94D41

01010000206A6900009ECA9A4D551D5041534CAB00C0CD4D41

01010000206A690000D573E4314C2450410F6A56C0C9C94D41

01010000206A690000A7CBF948D5185041DAAAB0F84BD24D41

01010000206A6900008A238B13151E50416DB1018FAFD74D41

01010000206A69000021722462B61F50414E6B761C38C14D41

01010000206A690000A72C6671721F50415FF5395DB6D54D41

01010000206A69000001E369250D2150411ACF92E637D24D41

01010000206A690000F48DA66C572750419BA03F55B0C64D41

01010000206A69000062576097721A504155AA9F8D12D54D41

01010000206A69000069E5BB9C412250410F7689C054BB4D41

01010000206A6900006795F01D5C1A504182B3152145CE4D41

01010000206A690000949040228D1C50418B6D583ACCC24D41

01010000206A690000EDB9A38834235041CA8DBA1EBCBC4D41

01010000206A690000B69A06F4DD21504155045E6B09C24D41

01010000206A690000DEF0FA99831C5041605685812CC14D41

01010000206A690000E5FE60A3532550418AF500B85CCA4D41

01010000206A6900005B8526C2C61E504110C1105DAABC4D41

01010000206A69000073386DB112235041A6E983CFCCC34D41

01010000206A690000F1B262EA0B2450416577FAB859D04D41

01010000206A690000525A9DBFB61E50416FD5FAC373D44D41

01010000206A6900008A1B40FBD5255041CA656B310FCD4D41

01010000206A6900004DAD641A011C504184ABF5DCFAD84D41

01010000206A690000B085D2B1A6215041ADA2D0FE3FC44D41

01010000206A690000201C42620B295041CE3BD53F17C74D41

01010000206A690000FE7EDD7DE31D5041D1782A1809CB4D41

01010000206A690000AD618BBE2E245041BBB8DE5F30CD4D41

01010000206A69000026749310931D504163D1AD1B77D84D41

01010000206A690000F8F5273C4E1F5041E948C16067C64D41

01010000206A6900007902C783F924504128D51946B8C44D41

01010000206A6900001CCD78CBCD265041BC5778A7CBC54D41

01010000206A6900000DF4DE35E523504106A928F00FCA4D41

01010000206A690000E87D3EC6871B5041BF10484A7DCD4D41

01010000206A690000270076B9851F5041DEC400C7A1BE4D41

01010000206A6900001416DAD3D41C50415BE656B3ADCE4D41

01010000206A6900007AAF6D756D1B50410C6BD638BED74D41

01010000206A69000099CA1728842150417BA8DAE026D04D41

01010000206A690000A27BD57BF51F5041F7D3EB5D8FD94D41

01010000206A69000068F2A3261B1F5041F9EA481FDFDA4D41

01010000206A690000DD883952F61B5041C11BAFA58DC74D41

01010000206A69000023D2F478D92450419DEC17ACD9BD4D41

01010000206A6900007725FAC0E22250419946E67B45D34D41

01010000206A69000049B437E452215041E1581D85B7D64D41

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

01010000206A6900006A9809CB5E2450416D7847E7CCC14D41

01010000206A69000038621F422F2050411AFFCDD3E0C74D41

01010000206A690000291139F6722650410A247CB1E8C24D41

01010000206A690000BD4388D966205041D4DE951418D44D41

01010000206A690000C6E92CECC21C50413705AEC32DDB4D41

01010000206A690000FCB6AA3C73205041DE1B4D472CCF4D41

01010000206A6900000E9B808C681C50419D7888EA7CCA4D41

01010000206A69000020AD7813131D50410179F7E239C24D41

SELECT 101