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

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	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	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 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	Nulpotency 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:

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
linear halving generalization of Baer correspondence	class two Lie ring	3	3
cocycle halving generalization of Baer correspondence	class two Lie cring	17	14
cocycle skew reversal generalization of Baer correspondence	class two near-Lie cring	?	?
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	?	?

Here is a long version:

Non-abelian member of pair	GAP ID	Abelian member of pair	GAP ID	The 1-isomorphism arises as a ...	Best perspective 2
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
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
semidirect product of Z16 and Z4 via fifth power map	28	direct product of Z16 and Z4	26	?
	64	direct product of Z4 and Z4 and Z4	55	?
	82	direct product of Z4 and Z4 and Z4	55	?
	17	direct product of Z8 and Z4 and Z2	83	?
	25	direct product of Z8 and Z4 and Z2	83	?
SmallGroup(64,113)	113	direct product of Z8 and Z4 and Z2	83	?
SmallGroup(64,114)	114	direct product of Z8 and Z4 and Z2	83	?
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
	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	via class three Lie cring(?)
SmallGroup(64,210)	210	direct product of Z4 and Z4 and V4	192	?