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

167 groups of the catalog are in this class; the 139 with an abelian direct factor are hidden.

group	order	× stem
#2032	128	×1	◆ stem
#11089	256	×2
#11090	256	×2
#11279	256	×2
#11280	256	×2
#11394	256	×2
#11395	256	×2
#23474	256	×2
#115977	512	×4
#120727	512	×4
#121987	512	×4
#123574	512	×4
#123575	512	×4
#123776	512	×4
#123777	512	×4
#123778	512	×4
#123779	512	×4
#124007	512	×4
#124008	512	×4
#124009	512	×4
#124010	512	×4
#147972	512	×4
#148930	512	×4
#149422	512	×4
#357766	512	×4
#358293	512	×4
#358789	512	×4
#359177	512	×4

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.