Element structure of groups of order 64

Grouping by conjugacy class sizes
Here now is a grouping by conjugacy class sizes. Note that since number of conjugacy classes in group of prime power order is congruent to order of group modulo prime-square minus one, all the values for the number of conjugacy classes are congruent to 64 mod 3, and hence congruent to 1 mod 3.

Here is the GAP code to generate this:

We use the coded (not in-built) function ConjugacyClassSizeGroupingFull (follow link to get code):

gap> C := ConjugacyClassSizeGroupingFull(64);; gap> D := List(C,x -> [x[1],x[2],Length(x[2]),Set(List(x[2],i->NilpotencyClassOfGroup(SmallGroup(64,i))))]); [ [ [ [ 1, 2 ], [ 2, 1 ], [ 4, 5 ], [ 8, 5 ] ], [ 32, 33, 34, 35, 36, 37 ], 6, [ 4 ] ], [ [ [ 1, 2 ], [ 2, 3 ], [ 4, 8 ], [ 8, 3 ] ], [ 134, 135, 136, 137, 138, 139 ], 6, [ 3 ] ],  [ [ [ 1, 2 ], [ 2, 5 ], [ 4, 5 ], [ 8, 4 ] ], [ 41, 42, 43, 46, 152, 153, 154, 190, 191 ], 9, [ 3, 4 ] ],  [ [ [ 1, 2 ], [ 2, 9 ], [ 4, 11 ] ], [ 257, 258, 259 ], 3, [ 3 ] ], [ [ [ 1, 2 ], [ 2, 15 ], [ 16, 2 ] ], [ 52, 53, 54 ], 3, [ 5 ] ],  [ [ [ 1, 4 ], [ 2, 2 ], [ 4, 6 ], [ 8, 4 ] ], [ 149, 150, 151, 170, 171, 172, 177, 178, 182 ], 9, [ 3 ] ],  [ [ [ 1, 4 ], [ 2, 4 ], [ 4, 9 ], [ 8, 2 ] ], [ 8, 9, 10, 11, 12, 13, 14, 128, 129, 130, 131, 132, 133, 140, 141, 142, 143, 144, 145, 155, 156, 157,          158, 159, 160, 161, 162, 163, 164, 165, 166 ], 31, [ 3 ] ],  [ [ [ 1, 4 ], [ 2, 6 ], [ 4, 12 ] ], [ 4, 5, 18, 19, 23, 24, 25, 28, 30, 90, 91, 92, 93, 94, 98, 99, 100, 102, 109, 111, 121, 122, 123, 125, 215, 216,          217, 218, 219, 220, 221, 222, 223, 224, 225, 254, 255, 256 ], 38, [ 2, 3 ] ],  [ [ [ 1, 4 ], [ 2, 12 ], [ 4, 9 ] ], [ 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240 ], 15, [ 2 ] ],  [ [ [ 1, 4 ], [ 2, 14 ], [ 8, 4 ] ], [ 38, 39, 40, 47, 48, 49, 146, 147, 148, 167, 168, 169, 173, 174, 175, 176, 179, 180, 181, 186, 187, 188, 189 ],      23, [ 3, 4 ] ], [ [ [ 1, 4 ], [ 2, 30 ] ], [ 199, 200, 201, 249, 264, 265, 266 ], 7, [ 2 ] ],  [ [ [ 1, 4 ], [ 4, 15 ] ], [ 241, 242, 243, 244, 245 ], 5, [ 2 ] ],  [ [ [ 1, 8 ], [ 2, 12 ], [ 4, 8 ] ], [ 6, 7, 15, 16, 20, 21, 22, 31, 45, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 88, 89, 95, 96, 97, 101,          104, 105, 106, 107, 108, 110, 113, 114, 116, 117, 118, 119, 120, 124, 127, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214,          250, 251, 252, 253 ], 60, [ 2, 3 ] ], [ [ [ 1, 8 ], [ 4, 14 ] ], [ 73, 74, 75, 76, 77, 78, 79, 80, 81, 82 ], 10, [ 2 ] ],  [ [ [ 1, 16 ], [ 2, 24 ] ], [ 3, 17, 27, 29, 44, 51, 56, 57, 58, 59, 84, 85, 86, 87, 103, 112, 115, 126, 184, 185, 193, 194, 195, 196, 197, 198, 247,          248, 261, 262, 263 ], 31, [ 2 ] ], [ [ [ 1, 64 ] ], [ 1, 2, 26, 50, 55, 83, 183, 192, 246, 260, 267 ], 11, [ 1 ] ] ]

Grouping by cumulative conjugacy class sizes (number of elements)
Note that it is not true that the cumulative conjugacy class size statistics values divide the order of the group in all cases. There are a few counterexamples in the table above, as we can see values such as 12, 20, 28, and 40. $$p^6$$ is the smallest prime power where such examples exist. See also:


 * There exist groups of prime-sixth order in which the cumulative conjugacy class size statistics values do not divide the order of the group
 * All cumulative conjugacy class size statistics values divide the order of the group for groups up to prime-fifth order

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):

Here is a summary version:

A total of 23 of the 29 1-isomorphisms are explained using the explanations here. Here is a long version:

Grouping by abelian member
Of the 11 abelian groups of order 64, 9 are 1-isomorphic to non-abelian groups. The only two that aren't are cyclic group:Z64, on account of the fact that finite group having the same order statistics as a cyclic group is cyclic, and elementary abelian group:E64, on account of the fact that exponent two implies abelian.