Element structure of groups of order 64

This article gives specific information, namely, element structure, about a family of groups, namely: groups of order 64.
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Pairs where one of the groups is abelian

There are 29 pairs of groups that are 1-isomorphic with the property that one of them is abelian. Of these, some pairs share the abelian group part, as the table below shows. Of these, the only example of a group that is not of nilpotency class two is SmallGroup(64,25) (GAP ID: 25):

Non-abelian member of pair	GAP ID	Abelian member of pair	GAP ID	Type of the 1-isomorphism	Long explanation for the 1-isomorphism	Description of the 1-isomorphism	Best perspective 1	Best perspective 2	Alternative perspective
semidirect product of Z8 and Z8 of M-type	3	direct product of Z8 and Z8	2	via class two Lie cring
semidirect product of Z16 and Z4 of M-type	27	direct product of Z16 and Z4	26	via class two Lie cring
M64	51	direct product of Z32 and Z2	50	via class two Lie cring
SmallGroup(64,57)	57	direct product of Z4 and Z4 and Z4	55	via class two Lie cring
direct product of SmallGroup(32,4) and Z2	84	direct product of Z8 and Z4 and Z2	83	via class two Lie cring
direct product of M16 and Z4	85	direct product of Z8 and Z4 and Z2	83	via class two Lie cring
central product of M16 and Z8 over common Z2	86	direct product of Z8 and Z4 and Z2	83	via class two Lie cring
	112	direct product of Z8 and Z4 and Z2	83	via class two Lie cring
direct product of M32 and Z2	184	direct product of Z16 and V4	183	via class two Lie cring
central product of D8 and Z16	185	direct product of Z16 and V4	183	via class two Lie cring
direct product of SmallGroup(32,24) and Z2	195	direct product of Z4 and Z4 and V4	192	via class two Lie cring
direct product of SmallGroup(16,13) and Z4	198	direct product of Z4 and Z4 and V4	192	via class two Lie cring
direct product of M16 and V4	247	direct product of Z8 and E8	246	via class two Lie cring
SmallGroup(64,248)	248	direct product of Z8 and E8	246	via class two Lie cring
	249	direct product of Z8 and E8	246	via class two Lie cring
direct product of SmallGroup(16,13) and V4	263	direct product of E16 and Z4	260	via class two Lie cring
	266	direct product of E16 and Z4	260	via class two Lie cring
semidirect product of Z16 and Z4 via fifth power map	28	direct product of Z16 and Z4	26	?
	64	direct product of Z4 and Z4 and Z4	55	?
	82	direct product of Z4 and Z4 and Z4	55	?
	17	direct product of Z8 and Z4 and Z2	83	?
	25	direct product of Z8 and Z4 and Z2	83	?
	113	direct product of Z8 and Z4 and Z2	83	?
	114	direct product of Z8 and Z4 and Z2	83	?
direct product of SmallGroup(32,2) and Z2	56	direct product of Z4 and Z4 and V4	192	via class two near-Lie cring
	61	direct product of Z4 and Z4 and V4	192	?
	77	direct product of Z4 and Z4 and V4	192	?
direct product of SmallGroup(32,33) and Z2	209	direct product of Z4 and Z4 and V4	192	via class three Lie cring
	210	direct product of Z4 and Z4 and V4	192	?

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