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 ◆.

279 groups of the catalog are in this class; the 230 with an abelian direct factor are hidden.

group	order	× stem
#2180	128	×1	◆ stem
AGL(1,ℤ/16)	128	×1	◆ stem
#12065	256	×2
#12066	256	×2
#12067	256	×2
#12068	256	×2
#12647	256	×2
#12648	256	×2
#12649	256	×2
#12650	256	×2
#12651	256	×2
#12677	256	×2
#12678	256	×2
#33468	256	×2
#122961	512	×4
#122962	512	×4
#122963	512	×4
#128554	512	×4
#128555	512	×4
#128556	512	×4
#128557	512	×4
#128688	512	×4
#128689	512	×4
#128690	512	×4
#128691	512	×4
#128692	512	×4
#128693	512	×4
#128694	512	×4
#128695	512	×4
#129120	512	×4
#129121	512	×4
#129164	512	×4
#129165	512	×4
#129166	512	×4
#129170	512	×4
#129171	512	×4
#146344	512	×4
#146345	512	×4
#146346	512	×4
#146347	512	×4
#151103	512	×4
#483589	512	×4
#483590	512	×4
#483591	512	×4
#483599	512	×4
#493077	512	×4
#493084	512	×4
#493158	512	×4
#493247	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.