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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Conjugacy class sizes

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

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.

Number of size 1 conjugacy classes	Number of size 2 conjugacy classes	Number of size 4 conjugacy classes	Number of size 8 conjugacy classes	Number of size 16 conjugacy classes	Total number of conjugacy classes	Total number of groups	Nilpotency class(es) attained by these groups	Hall-Senior family/families	List of GAP IDs (second part)
64	0	0	0	0	64	11	1	$\Gamma _{1}$, i.e., all the abelian groups of order 64	[SHOW MORE] 1, 2, 26, 50, 55, 83, 183, 192, 246, 260, 267
16	24	0	0	0	40	31	2	$\Gamma _{2}$	[SHOW MORE] 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
8	12	8	0	0	28	60	2,3	$\Gamma _{3}$ (class three) and $\Gamma _{4}$ (class two)	[SHOW MORE] 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
8	0	14	0	0	22	10	2	?	[SHOW MORE] 73, 74, 75, 76, 77, 78, 79, 80, 81, 82
4	30	0	0	0	34	7	2	$\Gamma _{5}$	[SHOW MORE] 199, 200, 201, 249, 264, 265, 266
4	14	0	4	0	22	23	3, 4	$\Gamma _{8}$	[SHOW MORE] 38, 39, 40, 47, 48, 49, 146, 147, 148, 167, 168, 169, 173, 174, 175, 176, 179, 180, 181, 186, 187, 188, 189
4	12	9	0	0	25	15	2		[SHOW MORE] 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240
4	6	12	0	0	22	38	2,3	$\Gamma _{6}$ and $\Gamma _{7}$	[SHOW MORE] 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
4	4	9	2	0	19	31	3		[SHOW MORE] 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
4	2	6	4	0	16	9	3		[SHOW MORE] 149, 150, 151, 170, 171, 172, 177, 178, 182
4	0	15	0	0	19	5	2		[SHOW MORE] 241, 242, 243, 244, 245
2	15	0	0	2	19	3	5		[SHOW MORE] 52, 53, 54
2	9	11	0	0	22	3	3		[SHOW MORE] 257, 258, 259
2	5	5	4	0	16	9	3,4		[SHOW MORE] 41, 42, 43, 46, 152, 153, 154, 190, 191
2	3	8	3	0	16	6	3		[SHOW MORE] 134, 135, 136, 137, 138, 139
2	1	5	5	0	13	6	4		[SHOW MORE] 32, 33, 34, 35, 36, 37

Here is the GAP code to generate this:[SHOW MORE]

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)

Number of elements in (size at most 1) conjugacy classes	Number of elements in (size at most 2) conjugacy classes	Number of elements in (size at most 4) conjugacy classes	Number of elements in (size at most 8) conjugacy classes	Number of elements in (size at most 16) conjugacy classes	Total number of conjugacy classes	Total number of groups	Nilpotency class(es) attained by these groups	Hall-Senior family/families	List of GAP IDs (second part)
64	64	64	64	64	64	11	1	$\Gamma _{1}$, i.e., all the abelian groups of order 64	[SHOW MORE] 1, 2, 26, 50, 55, 83, 183, 192, 246, 260, 267
16	64	64	64	64	40	31	2	$\Gamma _{2}$	[SHOW MORE] 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
8	32	64	64	64	28	60	2,3	$\Gamma _{3}$ (class three) and $\Gamma _{4}$ (class two)	[SHOW MORE] 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
8	8	64	64	64	22	10	2	?	[SHOW MORE] 73, 74, 75, 76, 77, 78, 79, 80, 81, 82
4	64	64	64	64	34	7	2	$\Gamma _{5}$	[SHOW MORE] 199, 200, 201, 249, 264, 265, 266
4	32	32	64	64	22	23	3, 4	$\Gamma _{8}$	[SHOW MORE] 38, 39, 40, 47, 48, 49, 146, 147, 148, 167, 168, 169, 173, 174, 175, 176, 179, 180, 181, 186, 187, 188, 189
4	28	64	64	64	25	15	2		[SHOW MORE] 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240
4	16	64	64	64	24	38	2,3	$\Gamma _{6}$ and $\Gamma _{7}$	[SHOW MORE] 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
4	12	48	64	64	19	31	3		[SHOW MORE] 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
4	8	32	64	64	16	9	3		[SHOW MORE] 149, 150, 151, 170, 171, 172, 177, 178, 182
4	4	64	64	64	19	5	2		[SHOW MORE] 241, 242, 243, 244, 245
2	32	32	32	64	19	3	5		[SHOW MORE] 52, 53, 54
2	20	64	64	64	22	3	3		[SHOW MORE] 257, 258, 259
2	12	32	64	64	16	9	3,4		[SHOW MORE] 41, 42, 43, 46, 152, 153, 154, 190, 191
2	8	40	64	64	16	6	3		[SHOW MORE] 134, 135, 136, 137, 138, 139
2	4	24	64	64	13	6	4		[SHOW MORE] 32, 33, 34, 35, 36, 37

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

1-isomorphism

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:

Nature of 1-isomorphism	Intermediate object	Number of 1-isomorphisms between non-abelian and abelian group of this type	Number of 1-isomorphisms between non-abelian and abelian group of this nature, not of any of the preceding types	Note
linear halving generalization of Baer correspondence	class two Lie ring	3	3	The examples are SmallGroup(64,57), direct product of SmallGroup(32,4) and Z2 (ID: (64,84)), and semidirect product of Z16 and Z4 of M-type (ID: (64,27)). See table below.
cocycle halving generalization of Baer correspondence	class two Lie cring	17	14	See table below.
cocycle skew reversal generalization of Baer correspondence	class two near-Lie cring	19	2	The two new examples are SmallGroup(64,17) and direct product of SmallGroup(32,2) and Z2.
linear halving generalization of Lazard correspondence	class three Lie ring	4	0	Of the four examples, three arise via the linear halving generalization of Baer correspondence. The fourth, namely semidirect product of Z8 and Z8 of M-type (ID: (64,3)), alternatively arises via the cocycle halving generalization of Baer correspondence.
cocycle halving generalization of Lazard correspondence	class three Lie cring	21	4	The new examples are semidirect product of Z16 and Z4 via fifth power map (ID: 28), SmallGroup(64,113), SmallGroup(64,114) and SmallGroup(64,210). See table below.

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

Non-abelian member of pair	GAP ID	Abelian member of pair	GAP ID	The 1-isomorphism arises as a ...	Description of the 1-isomorphism	Best perspective 1	Best perspective 2	Alternative perspective
semidirect product of Z16 and Z4 of M-type	27	direct product of Z16 and Z4	26	linear halving generalization of Baer correspondence, the intermediate object being a class two Lie ring			second cohomology group for trivial group action of direct product of Z4 and Z4 on Z4
SmallGroup(64,57)	57	direct product of Z4 and Z4 and Z4	55	linear halving generalization of Baer correspondence, the intermediate object being a class two Lie ring			second cohomology group for trivial group action of direct product of Z4 and Z4 on Z4
direct product of SmallGroup(32,4) and Z2	84	direct product of Z8 and Z4 and Z2	83	linear halving generalization of Baer correspondence, the intermediate object being a class two Lie ring			second cohomology group for trivial group action of direct product of Z4 and Z4 on Z4
semidirect product of Z8 and Z8 of M-type	3	direct product of Z8 and Z8	2	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring; alternatively, the linear halving generalization of Lazard correspondence, the intermediate object being a class three Lie ring
M64	51	direct product of Z32 and Z2	50	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
direct product of M16 and Z4	85	direct product of Z8 and Z4 and Z2	83	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
central product of M16 and Z8 over common Z2	86	direct product of Z8 and Z4 and Z2	83	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
	112	direct product of Z8 and Z4 and Z2	83	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
direct product of M32 and Z2	184	direct product of Z16 and V4	183	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
central product of D8 and Z16	185	direct product of Z16 and V4	183	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
direct product of SmallGroup(32,24) and Z2	195	direct product of Z4 and Z4 and V4	192	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
direct product of SmallGroup(16,13) and Z4	198	direct product of Z4 and Z4 and V4	192	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
direct product of M16 and V4	247	direct product of Z8 and E8	246	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
SmallGroup(64,248)	248	direct product of Z8 and E8	246	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
	249	direct product of Z8 and E8	246	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
direct product of SmallGroup(16,13) and V4	263	direct product of E16 and Z4	260	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
	266	direct product of E16 and Z4	260	cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
SmallGroup(64,17)	17	direct product of Z8 and Z4 and Z2	83	cocycle skew reversal generalization of Baer correspondence, the intermediate object being a class two near-Lie cring
direct product of SmallGroup(32,2) and Z2	56	direct product of Z4 and Z4 and V4	192	cocycle skew reversal generalization of Baer correspondence, the intermediate object being a class two near-Lie cring
semidirect product of Z16 and Z4 via fifth power map	28	direct product of Z16 and Z4	26	cocycle halving generalization of Lazard correspondence, the intermediate object being a class three Lie cring
SmallGroup(64,113)	113	direct product of Z8 and Z4 and Z2	83	cocycle halving generalization of Lazard correspondence, the intermediate object being a class three Lie cring
SmallGroup(64,114)	114	direct product of Z8 and Z4 and Z2	83	cocycle halving generalization of Lazard correspondence, the intermediate object being a class three Lie cring
SmallGroup(64,210)	210	direct product of Z4 and Z4 and V4	192	cocycle halving generalization of Lazard correspondence, the intermediate object being a class three Lie cring
	64	direct product of Z4 and Z4 and Z4	55	?
	82	direct product of Z4 and Z4 and Z4	55	?
SmallGroup(64,25)	25	direct product of Z8 and Z4 and Z2	83	?
	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

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.

Abelian member	GAP ID	Total number of members (excluding abelian member)	Other members	GAP IDs (in order of listing)	Hall-Senior symbols (in order of listing)	Hall-Senior numbers (in order of listing)
direct product of Z8 and Z8	2	1	semidirect product of Z8 and Z8 of M-type	3
direct product of Z16 and Z4	26	2	semidirect product of Z16 and Z4 of M-type, semidirect product of Z16 and Z4 via fifth power map	27, 28
direct product of Z32 and Z2	50	1	M64	51
direct product of Z4 and Z4 and Z4	55	3	SmallGroup(64,57), SmallGroup(64,64), SmallGroup(64,82)	57, 64, 82
direct product of Z8 and Z4 and Z2	83	8	[SHOW MORE] SmallGroup(64,17), SmallGroup(64,25), direct product of SmallGroup(32,4) and Z2, direct product of M16 and Z4, central product of M16 and Z8 over common Z2, SmallGroup(64,112), SmallGroup(64,113), SmallGroup(64,114)	17, 25, 84, 85, 86, 112, 113, 114
direct product of Z16 and V4	183	2	direct product of M32 and Z2, central product of D8 and Z16	184, 185
direct product of Z4 and Z4 and V4	192	7	[SHOW MORE] direct product of SmallGroup(32,2) and Z2, SmallGroup(64,61), SmallGroup(64,77), direct product of SmallGroup(32,24) and Z2, direct product of SmallGroup(16,13) and Z4, SmallGroup(64,209), SmallGroup(64,210)	56, 61, 77, 195, 198, 209, 210
direct product of Z8 and E8	246	3	direct product of M16 and V4, SmallGroup(64,248), SmallGroup(64,249)	247, 248, 249
direct product of E16 and Z4	260	2	direct product of SmallGroup(16,13) and V4, SmallGroup(64,266)	263, 266
Total	--	29	--	--	--	--