MySQL 8.0

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CREATE TABLE tbl_Participants (id int, group_name varchar(10), firstname varchar(20), lastname  varchar(20));

INSERT INTO tbl_Participants (id, group_name, firstname, lastname) VALUES
(1, 'group1', 'firstname1', 'lastname1'),
(2, 'group1', 'firstname2', 'lastname2'),
(3, 'group1', 'firstname3', 'lastname3'),
(4, 'group1', 'firstname4', 'lastname4'),
(5, 'group2', 'firstname5', 'lastname5'),
(6, 'group2', 'firstname6', 'lastname6'),
(5, 'group2', 'firstname7', 'lastname7'),
(6, 'group3', 'firstname8', 'lastname8');
SELECT group_name,
       MAX(CASE WHEN rn = 1 THEN name END) Member1,
       MAX(CASE WHEN rn = 2 THEN name END) Member2,
       MAX(CASE WHEN rn = 3 THEN name END) Member3
FROM (
  SELECT group_name, 
         CONCAT(lastname, ' ', firstname) name,
         ROW_NUMBER() OVER (PARTITION BY group_name ORDER BY lastname, firstname) rn
  FROM tbl_Participants
) t
WHERE rn <= 3
GROUP BY group_name


group_name
Member1
Member2
Member3


group1
lastname1 firstname1
lastname2 firstname2
lastname3 firstname3

group2
lastname5 firstname5
lastname6 firstname6
lastname7 firstname7

group3
lastname8 firstname8
null
null


SELECT group_name,
       SUBSTRING_INDEX(names, ',', 1) Member1,
       CASE WHEN counter > 1 THEN SUBSTRING_INDEX(SUBSTRING_INDEX(names, ',', 2), ',', -1) END Member2,
       CASE WHEN counter > 2 THEN SUBSTRING_INDEX(SUBSTRING_INDEX(names, ',', 3), ',', -1) END Member3
FROM (
  SELECT group_name, 
         COUNT(*) counter,
         GROUP_CONCAT(CONCAT(lastname, ' ', firstname) ORDER BY lastname, firstname) names
  FROM tbl_Participants
  GROUP BY group_name
) t  


group_name
Member1
Member2
Member3


group1
lastname1 firstname1
lastname2 firstname2
lastname3 firstname3

group2
lastname5 firstname5
lastname6 firstname6
lastname7 firstname7

group3
lastname8 firstname8
null
null

group_name	Member1	Member2	Member3
group1	lastname1 firstname1	lastname2 firstname2	lastname3 firstname3
group2	lastname5 firstname5	lastname6 firstname6	lastname7 firstname7
group3	lastname8 firstname8	null	null