# 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 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]

## 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