Groups of order 2^n
Contents
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 | Value | Number of groups of order | 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 | Value | Number of groups of order | 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 | Value | Number of groups of order | 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:
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:
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 , see arithmetic functions for groups of order 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.
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 |
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