Groups of order 2^n

Number of groups of small orders

FACTS TO CHECK AGAINST (number of groups of prime power order):
Exact counts: Higman-Sims asymptotic formula on number of groups of prime power order |
Upper and lower bounds: Higman-Sims asymptotic formula on number of groups of prime power order (best known) | upper bound on number of groups of prime power order using power-commutator presentations (very crude) | inductive upper bound on number of groups of prime power order using power-commutator presentations (very crude)

Exponent $n$	Value $2^n$	Number of groups of order $2^n$	Greatest integer function of logarithm of number of groups to base 2	Reason/Explanation/List	Historical point of first classification
0	1	1	0	Only trivial group	--
1	2	1	0	Only cyclic group:Z2; see equivalence of definitions of group of prime order	--
2	4	2	1	Only cyclic group:Z4 and Klein four-group; see also groups of order 4 and classification of groups of prime-square order	Before 1890
3	8	5	2	See groups of order 8; see classification of groups of prime-cube order	Before 1890
4	16	14	3	See groups of order 16	Before 1890
5	32	51	5	See groups of order 32	Published 1896 in a paper by G. A. Miller
6	64	267	8	See groups of order 64
7	128	2328	11	See groups of order 128	Published 1990 in a paper by James, Newman, and O'Brien
8	256	56092	15	See groups of order 256	Published 1991 in a paper by O'Brien
9	512	10494213	23	See groups of order 512	A counting without an explicit listing of the groups themselves was first reported in a 1997 conference by Eick and O'Brien and the methods were elaborated upon in a paper published in 1999.
10	1024	49487365422	35	See groups of order 1024
11	2048	unknown	unknown	See groups of order 2048
12	4096	unknown	unknown	See groups of order 4096

Counts for various equivalence classes

Up to isoclinism

FACTS TO CHECK AGAINST for isoclinic groups (groups with an isoclinism between them):
by definition, isoclinic groups have isomorphic inner automorphism groups and isomorphic derived subgroups, with the isomorphisms compatible.
isoclinic groups have same nilpotency class|isoclinic groups have same derived length | isoclinic groups have same proportions of conjugacy class sizes | isoclinic groups have same proportions of degrees of irreducible representations

Exponent $n$	Value $2^n$	Number of groups of order $2^n$	Number of equivalence classes under isoclinism (these equivalence classes are also called Hall-Senior families)	Sizes of equivalence classes (i.e., number of groups in each Hall-Senior family)	Additional note
0	1	1	1	1	trivial group only
1	2	1	1	1	cyclic group:Z2 only
2	4	2	1	2	both groups are abelian groups
3	8	5	2	3, 2	abelian groups form one equivalence class, the non-abelian groups form another. See groups of order 8#Families and classification.
4	16	14	3	5, 6, 3	One equivalence class for each nilpotency class value. Class one (abelian groups), class two, and class three. See groups of order 16#Families and classification.
5	32	51	8	7, 15, 10, 9, 2, 2, 3, 3	See groups of order 32#Families and classification and also classification of groups of order 32.
6	64	267	27		See groups of order 64#Families and classification and also classification of groups of order 64.
7	128	2328	115		See groups of order 128#Families and classification and also classification of groups of order 128.
8	256	56092	?

Up to isologism

Exponent $n$	Value $2^n$	Number of groups of order $2^n$	Number of equivalence classes under isoclinism (i.e., isologism for class one)	Number of equivalence classes under isologism for class two	Number of equivalence classes under isologism for class three	Number of equivalence classes under isologism for class four	Number of equivalence classes under isologism for class five
0	1	1	1	1	1	1	1
1	2	1	1	1	1	1	1
2	4	2	1	1	1	1	1
3	8	5	2	1	1	1	1
4	16	14	3	2	1	1	1
5	32	51	8	3	2	1	1
6	64	267	27	?	?	2	1
7	128	2328	115	?	?	?	2
8	256	56092	?	?	?	?	?

Arithmetic functions

Further information: Arithmetic functions for groups of order 2^n

Summary

Below is a summary of the behavior of the average values for important arithmetic functions, where the average is computed by equally weighting all isomorphism classes of groups of that order:

$n$	$2^n$	number of groups	nilpotency class	derived length	prime-base logarithm of exponent	Frattini length	minimum size of generating set
0	1	1	0	0	0	0	0
1	2	1	1	1	1	1	1
2	4	2	1	1	1.5	1.5	1.5
3	8	5	1.4	1.4	2	2	2
4	16	14	1.8571	1.6429	2.4286	2.4286	2.3571
5	32	51	2.2745	1.8627	2.6471	2.6471	2.7451
6	64	267	2.5843	1.9588	2.7828	2.7903	3.1124
7	128	2328	2.6937	1.9996	2.7637	2.7668	3.5833
8	256	56092	2.4941	2.0069	?	?	?

Below is a summary of the behavior of the average values where the groups are weighted by the cohomology tree probability distribution:

$n$	$2^n$	number of groups	nilpotency class	derived length	prime-base logarithm of exponent	Frattini length	minimum size of generating set
0	1	1	0	0	0	0	0
1	2	1	1	1	1	1	1
2	4	2	1	1	1.5	1.5	1.5
3	8	5	1.25	1.25	2.1875	?	1.8125
4	16	14	1.6172	1.5234	2.6865	?	2.0322
5	32	51	1.9889	1.7728	3.1426	?	2.2039
6	64	267	2.3329	1.8467	3.5342	?	2.3288

Details

We provide information on nilpotency class below. For more details on the distribution of values of other arithmetic functions for groups of order $2^n$, see arithmetic functions for groups of order 2^n.

$n$	$2^n$	total number of groups	class 0	class 1	class 2	class 3	class 4	class 5	class 6	class 7
0	1	1	1
1	2	1	0	1
2	4	2	0	2
3	8	5	0	3	2
4	16	14	0	5	6	3
5	32	51	0	7	26	15	3
6	64	267	0	11	117	114	22	3
7	128	2328	0	15	947	1137	197	29	3
8	256	56092	0	22	31742	21325	2642	320	38	3

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

We use the function SortArithmeticFunctionSizes, which is not in-built but is easy to code (follow link to get code). We also use the in-built function NilpotencyClassOfGroup. Using these functions, the above data can be generated as follows:

gap> SortArithmeticFunctionSizes(2,0,NilpotencyClassOfGroup);
[ [ 0, 1 ] ]
gap> SortArithmeticFunctionSizes(2,1,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 1 ] ]
gap> SortArithmeticFunctionSizes(2,2,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ]
gap> SortArithmeticFunctionSizes(2,3,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ]
gap> SortArithmeticFunctionSizes(2,4,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 5 ], [ 2, 6 ], [ 3, 3 ], [ 4, 0 ] ]
gap> SortArithmeticFunctionSizes(2,5,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 26 ], [ 3, 15 ], [ 4, 3 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(2,6,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 11 ], [ 2, 117 ], [ 3, 114 ], [ 4, 22 ], [ 5, 3 ], [ 6, 0 ] ]
gap> SortArithmeticFunctionSizes(2,7,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 15 ], [ 2, 947 ], [ 3, 1137 ], [ 4, 197 ], [ 5, 29 ], [ 6, 3 ], [ 7, 0 ] ]
gap> SortArithmeticFunctionSizes(2,8,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 22 ], [ 2, 31742 ], [ 3, 21325 ], [ 4, 2642 ], [ 5, 320 ], [ 6, 38 ], [ 7, 3 ], [ 8, 0 ] ]

Although the command can in principle be run for $2^8$, it requires a huge memory allocation (this can be controlled setting the "-o" command line option when starting GAP). To obtain the result for $2^8$, it is best to split the list of all small groups of order $2^8$ into sub-lists and run the function on each.

Here is the same information, now given in terms of the fraction of groups of a given order that are of a given nilpotency class. For ease of comparison, all fractions are written as decimals, rounded to the fourth decimal place.

$n$	$2^n$	total number of groups	average of values (equal weighting on all groups)	class 0	class 1	class 2	class 3	class 4	class 5	class 6	class 7
0	1	1	0	1
1	2	1	1	0	1
2	4	2	1	0	1
3	8	5	1.4	0	0.6000	0.4000
4	16	14	1.8571	0	0.3571	0.4286	0.2143
5	32	51	2.2745	0	0.1373	0.5098	0.2941	0.0588
6	64	267	2.5843	0	0.0412	0.4382	0.4270	0.0824	0.0112
7	128	2328	2.6937	0	0.0064	0.4068	0.4884	0.0846	0.01245	0.0013
8	256	56092	2.4941	0	0.0004	0.5659	0.3802	0.0471	0.0057	0.0007	0.0001

Below is the information for the probability distribution by nilpotency class using the cohomology tree probability distribution: [SHOW MORE]

$n$	$2^n$	total number of groups	weighted average nilpotency class under the distribution	class 0	class 1	class 2	class 3	class 4	class 5	class 6	class 7
0	1	1	0	1
1	2	1	1	0	1
2	4	2	1	0	1
3	8	5	1.25	0	0.7500	0.2500
4	16	14	1.6172	0	0.4766	0.4297	0.0938
5	32	51	1.9889	0	0.2772	0.4684	0.2427	0.0117
6	64	267	2.3329	0	0.1533	0.4231	0.3623	0.0598	0.0015

Elements

Further information: element structure of groups of order 2^n

Linear representation theory

Further information: linear representation theory of groups of order 2^n

Subgroups

Further information: subgroup structure of groups of order 2^n