Isoclinism class

Groups whose commutator structure agrees: there are isomorphisms G/Z(G) → H/Z(H) and G′ → H′ carrying one commutator map to the other. The class spans orders — G × A is isoclinic to G for every abelian A.

Stem order 81: every member's order is a multiple of it, and the members attaining it — the stem groups — are marked ◆.

661 groups of the catalog are in this class; the 611 with an abelian direct factor are hidden.

group	order	× stem
AΣL(1,𝔽₂₇)	81	×1	◆ stem
#731	81	×1	◆ stem
#732	81	×1	◆ stem
#733	81	×1	◆ stem
#6866	243	×3
#6867	243	×3
#6868	243	×3
#6869	243	×3
H(3,𝔽₃)⋊C₉	243	×3
3₋¹⁺²⋊C₉	243	×3
#6872	243	×3
#6873	243	×3
#6908	243	×3
#10624444	729	×9
#10624445	729	×9
#10624446	729	×9
H(3,𝔽₃)⋊C₂₇	729	×9
3₋¹⁺²⋊C₂₇	729	×9
#10624466	729	×9
#10624467	729	×9
#10624468	729	×9
#10624469	729	×9
#10624502	729	×9
#10624503	729	×9
#10624504	729	×9
#10624688	729	×9
#10624697	729	×9
#10624724	729	×9
#14546150	2,187	×27
#14546151	2,187	×27
#14546152	2,187	×27
#14546153	2,187	×27
#14546154	2,187	×27
#14546224	2,187	×27
#14546234	2,187	×27
#14546243	2,187	×27
H(3,𝔽₃)⋊C₈₁	2,187	×27
3₋¹⁺²⋊C₈₁	2,187	×27
#14546338	2,187	×27
#14546340	2,187	×27
#14546342	2,187	×27
#14546457	2,187	×27
#14546458	2,187	×27
#14546459	2,187	×27
#14546564	2,187	×27
#14546591	2,187	×27
#14550220	2,187	×27
#14550253	2,187	×27
#14550263	2,187	×27
#14551541	2,187	×27

The class itself is infinite — G × A is isoclinic to G for every abelian A, and there are abelian groups of every order — so a table shows the part of it the catalog names, never the whole.