Element structure of groups of order 81

Pairs where one of the groups is abelian
Of the 15 groups of order 81, 5 are abelian, 6 have nilpotency class two, and 4 have nilpotency class three. Via the Baer correspondence, each of the groups of class two has a Baer Lie ring, and in particular is 1-isomorphic to the additive group of that Lie ring. Of the 4 groups of nilpotency class three, only one (SmallGroup(81,8)) is 1-isomorphic to an abelian group. There are no 1-isomorphisms between pairs where both members are non-abelian.

Groupings that do not have any abelian members
There are no 1-isomorphisms between pairs where both members are non-abelian for order 81. Thus, there are no such groupings.

Order statistics raw data
Here is the GAP code to generate these order statistics:

We first define the OrderStatistics function (click through for the function code). Using this, the order statistics can be generated by:

List([1..15],i ->[OrderStatistics(SmallGroup(81,i)),i]);

GAP's output is:

[ [ [ 1, 2, 6, 18, 54 ], 1 ], [ [ 1, 8, 72, 0, 0 ], 2 ], [ [ 1, 26, 54, 0, 0 ], 3 ], [ [ 1, 8, 72, 0, 0 ], 4 ], [ [ 1, 8, 18, 54, 0 ], 5 ], [ [ 1, 8, 18, 54, 0 ], 6 ], [ [ 1, 44, 36, 0, 0 ], 7 ], [ [ 1, 26, 54, 0, 0 ], 8 ], [ [ 1, 62, 18, 0, 0 ], 9 ], [ [ 1, 8, 72, 0, 0 ], 10 ],  [ [ 1, 26, 54, 0, 0 ], 11 ], [ [ 1, 80, 0, 0, 0 ], 12 ], [ [ 1, 26, 54, 0, 0 ], 13 ], [ [ 1, 26, 54, 0, 0 ], 14 ], [ [ 1, 80, 0, 0, 0 ], 15 ] ]

Here are the cumulative order statistics.

Here is the GAP code to generate these cumulative order statistics:

List([1..15],i ->[OrderStatisticsCumulative(SmallGroup(81,i)),i]);

after first defining the OrderStatisticsCumulative function (follow link for GAP code for function definition).

Here is GAP's output:

[ [ [ 1, 3, 9, 27, 81 ], 1 ], [ [ 1, 9, 81, 81, 81 ], 2 ], [ [ 1, 27, 81, 81, 81 ], 3 ], [ [ 1, 9, 81, 81, 81 ], 4 ],  [ [ 1, 9, 27, 81, 81 ], 5 ], [ [ 1, 9, 27, 81, 81 ], 6 ],  [ [ 1, 45, 81, 81, 81 ], 7 ], [ [ 1, 27, 81, 81, 81 ], 8 ],  [ [ 1, 63, 81, 81, 81 ], 9 ], [ [ 1, 9, 81, 81, 81 ], 10 ],  [ [ 1, 27, 81, 81, 81 ], 11 ], [ [ 1, 81, 81, 81, 81 ], 12 ],  [ [ 1, 27, 81, 81, 81 ], 13 ], [ [ 1, 27, 81, 81, 81 ], 14 ],  [ [ 1, 81, 81, 81, 81 ], 15 ] ]

Equivalence classes based on order statistics
Here, we discuss the equivalence classes of groups of order 81 up to being order statistics-equivalent finite groups and up to the stronger notion of being 1-isomorphic groups (which means there is a bijection that restricts to isomorphisms on cyclic subgroups). See also order statistics-equivalent not implies 1-isomorphic.