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:

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
semidirect product of Z16 and Z4 via fifth power map	28	direct product of Z16 and Z4	26	?
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