Element structure of groups of order 2^n

This article describes the element structure of groups of order 2^n, i.e., groups whose order is a power of 2.

All groupings
We say that two groups are 1-isomorphic groups if there exists a bijection between them that restricts to an isomorphism on all cyclic subgroups on both sides, i.e., there exists a 1-isomorphism between them. Below, we briefly describe the equivalence classes of groups of order $$2^n$$ up to 1-isomorphism:

Pairs of abelian and non-abelian based on explanation
We consider pairs of the form (abelian group, non-abelian group) that are 1-isomorphic. Note that since finite abelian groups with the same order statistics are isomorphic, the non-abelian member cannot be shared between pairs. However, the abelian group could be shared between multiple pairs, i.e., a single abelian group could be 1-isomorphic to multiple non-abelian groups.

The table below provides information on the number of such pairs that can be explained on the basis of various correspondences. As we go from left to right, the correspondence becomes progressively more generic, so the numbers in the later columns include the groups satisfying the stronger, more specific correspondences from earlier columns (the exception being the linear halving generalization of Lazard correspondence column, which is not more generic than its preceding column).