Number of groups of given order

Definition

Let $n$ be a natural number. The number of groups of order $n$ is defined as the number of isomorphism classes of groups whose order is $n$.

This is a finite number and is bounded by $n^{n^2}$ for obvious reasons. The function is not strictly increasing in $n$ and depends heavily on the nature of the prime factorization of $n$.

Initial values

The ID of the sequence of these numbers in the Online Encyclopedia of Integer Sequences is A000001

Numbers up till 100

$n$	Number of groups of order $n$	Reason/explanation
1	1
2	1	prime number
3	1	prime number
4	2	square of a prime; see classification of groups of prime-square order
5	1	prime number
6	2	form $pq$ where $p,q$ primes, $q \mid p - 1$
7	1	prime number
8	5	prime cube: classification of groups of prime-cube order, also see groups of order 8
9	2	prime square; see classification of groups of prime-square order

Orders 10 to 36. We omit the prime numbers since there is only one group of each such order.[SHOW MORE]

$n$	Number of groups of order $n$	Reason/explanation
10	2	form $pq$ where $p,q$ primes, $q \mid p - 1$
12	5	see groups of order 12
14	2	form $pq$ where $p,q$ primes, $q \mid p - 1$
15	1	form $pq$ ($p,q$ primes) where $p$ doesn't divide $q - 1$, $q$ doesn't divide $p - 1$
16	14	see groups of order 16
18	5	see groups of order 18
20	5	see groups of order 20
21	2	form $pq$ where $p,q$ primes, $q \mid p - 1$
22	2	form $pq$ where $p,q$ primes, $q \mid p - 1$
24	15	see groups of order 24
25	2	prime square; see classification of groups of prime-square order
26	2	form $pq$ where $p,q$ primes, $q \mid p - 1$
27	5	see classification of groups of prime-cube order
28	4	see groups of order 28
30	4	see groups of order 30
32	51	see groups of order 32
33	1	form $pq$ ($p,q$ primes) where $p$ doesn't divide $q - 1$, $q$ doesn't divide $p - 1$
34	2	form $pq$ where $p,q$ primes, $q \mid p - 1$
35	1	form $pq$ ($p,q$ primes) where $p$ doesn't divide $q - 1$, $q$ doesn't divide $p - 1$
36	14	see groups of order 36

Orders greater than 36. We omit prime numbers, squares of primes, and numbers of the form $pq$ where $p,q$ both primes, since these are covered by standard cases.[SHOW MORE]

$n$	Number of groups of order $n$	Reason/explanation
40	14	see groups of order 40
42	6	see groups of order 42
44	4	see groups of order 44
45	2	both Sylow subgroups are normal, so it is a direct product.
48	52	see groups of order 48
50	5	see groups of order 50
52	5	see groups of order 52

Small powers of small primes

(For general formulas, see the next section).

Powers of 2:[SHOW MORE]

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

Further information: groups of order 2^n

Powers of 3: [SHOW MORE]

Exponent $n$	Value $3^n$	Number of groups of order $3^n$	Greatest integer function of logarithm of number of groups to base 3	Reason/Explanation/List
0	1	1	0	only trivial group
1	3	1	0	only cyclic group:Z3; see equivalence of definitions of group of prime order
2	9	2	0	cyclic group:Z9 and elementary abelian group:E9; see groups of order 9; see also classification of groups of prime-square order
3	27	5	1	see groups of order 27; see also classification of groups of prime-cube order
4	81	15	2	see groups of order 81; see also classification of groups of prime-fourth order for odd prime
5	243	67	3	see groups of order 243
6	729	504	5	see groups of order 729
7	2187	9310	8	see groups of order 2187
8	6561	unknown	unknown	see groups of order 6561
9	19683	unknown	unknown	see groups of order 19683

Further information: groups of order 3^n

Powers of 5: [SHOW MORE]

Exponent $n$	Value $5^n$	Number of groups of order $5^n$	List, information	Comparison with other primes, i.e., groups of order $p^n$
1	5	1	only cyclic group:Z5; see equivalence of definitions of group of prime order	See group of prime order
2	25	2	cyclic group:Z25 and elementary abelian group:E25; see also groups of order 25	groups of prime-square order, classification of groups of prime-square order
3	125	5	groups of order 125	groups of prime-cube order, classification of groups of prime-cube order
4	625	15	groups of order 625	groups of prime-fourth order, classification of groups of prime-fourth order for odd prime
5	3125	77	groups of order 3125	groups of prime-fifth order, see also Higman's PORC conjecture
6	15625	684	groups of order 15625	groups of prime-sixth order, see also Higman's PORC conjecture
7	78125	34297	groups of order 78125	groups of prime-seventh order, see also Higman's PORC conjecture

Further information: groups of order 5^n

Powers of 7: [SHOW MORE]

Exponent $n$	Value $57^n$	Number of groups of order $7^n$	Reason/Explanation/List
1	7	1	only cyclic group:Z7; see equivalence of definitions of group of prime order
2	49	2	cyclic group:Z49 and elementary abelian group:E49; see also groups of order 49 and classification of groups of prime-square order
3	343	5	see groups of order 343 and classification of groups of prime-cube order
4	2401	15	see groups of order 2401 and classification of groups of prime-fourth order for odd prime
5	16807	83	see groups of order 16807 and also the PORC formula in the table in the next section.
6	117649	860	see groups of order 117649 and also the PORC formula in the table in the next section.
7	823543	113147	see groups of order 823453

Arithmetic functions

Nilpotency class

$n$	$7^n$	total number of groups	class 0	class 1	class 2	class 3	class 4	class 5	class 6
0	1	1	1
1	7	1	0	1
2	49	2	0	2
3	343	5	0	3	2
4	2401	15	0	5	6	4
5	16807	83	0	7	32	33	11
6	117649	860	0	11	165	508	133	43

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(7,0,NilpotencyClassOfGroup);
[ [ 0, 1 ] ]
gap> SortArithmeticFunctionSizes(7,1,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 1 ] ]
gap> SortArithmeticFunctionSizes(7,2,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ]
gap> SortArithmeticFunctionSizes(7,3,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ]
gap> SortArithmeticFunctionSizes(7,4,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 5 ], [ 2, 6 ], [ 3, 4 ], [ 4, 0 ] ]
gap> SortArithmeticFunctionSizes(7,5,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 32 ], [ 3, 33 ], [ 4, 11 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(7,6,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 11 ], [ 2, 165 ], [ 3, 508 ], [ 4, 133 ], [ 5, 43 ], [ 6, 0 ] ]

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.

$n$	$7^n$	total number of groups	class 0	class 1	class 2	class 3	class 4	class 5	class 6
0	1	1	1
1	7	1	0	1
2	49	2	0	1
3	343	5	0	0.6000	0.4000
4	2401	15	0	0.3333	0.4000	0.2667
5	16807	83	0	0.0843	0.3855	0.3976	0.1325
6	117649	860	0	0.0128	0.1919	0.5907	0.1547	0.0500

Derived length

$n$	$7^n$	total number of groups	length 0	length 1	length 2	length 3
0	1	1	1
1	7	1	0	1
2	49	2	0	2
3	343	5	0	3	2
4	2401	15	0	5	10
5	16807	83	0	7	76
6	117649	860	0	11	829	20

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 DerivedLength. Using these functions, the above data can be generated as follows:

gap> SortArithmeticFunctionSizes(7,1,DerivedLength);
[ [ 0, 0 ], [ 1, 1 ] ]
gap> SortArithmeticFunctionSizes(7,2,DerivedLength);
[ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ]
gap> SortArithmeticFunctionSizes(7,3,DerivedLength);
[ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ]
gap> SortArithmeticFunctionSizes(7,4,DerivedLength);
[ [ 0, 0 ], [ 1, 5 ], [ 2, 10 ], [ 3, 0 ], [ 4, 0 ] ]
gap> SortArithmeticFunctionSizes(7,5,DerivedLength);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 76 ], [ 3, 0 ], [ 4, 0 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(7,6,DerivedLength);
[ [ 0, 0 ], [ 1, 11 ], [ 2, 829 ], [ 3, 20 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ] ]

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

$n$	$7^n$	total number of groups	length 0	length 1	length 2	length 3
0	1	1	1
1	7	1	0	1
2	49	2	0	1
3	343	5	0	0.6000	0.4000
4	2401	15	0	0.3333	0.6667
5	16807	83	0	0.0843	0.9157
6	117649	860	0	0.0128	0.9640	0.0233

Further information: groups of order 7^n

Facts

Basic facts

Value of $n$	What we can say about the number of groups of order $n$	Explanation
1	1	only the trivial group
$p$ a prime number	1	only the group of prime order. See equivalence of definitions of group of prime order
$p^2$, $p$ prime	2	only the cyclic group of prime-square order and the elementary abelian group of prime-square order
$p^3$, $p$ prime	5	see classification of groups of prime-cube order
$2^4 = 16$	14	see classification of groups of order 16, also groups of order 16 for summary information.
$p^4$, $p$ odd prime	15	see classification of groups of prime-fourth order for odd prime
$2^5 = 32$	51
$3^5 = 243$	67
$p^5$, prime $p \ge 5$	$2p + 61 + 2\operatorname{gcd}(p-1,3) + \operatorname{gcd}(p-1,4)$
$p^6$, prime $p \ge 5$	$3p^2 + 39p + 344 + 24 \operatorname{gcd}(p - 1,3) + 11 \operatorname{gcd}(p-1,4) + 2 \operatorname{gcd}(p-1,5)$
product $p_1p_2 \dots p_n$, $p_i$ distinct primes with no $p_i$ dividing $p_j - 1$	1	the cyclic group of that order. See classification of cyclicity-forcing numbers
product $pq$, $p,q$ primes with $p$ dividing $q - 1$	2
product $4p$, $p$ prime, $p > 3$, $p \equiv -1 \pmod 4$	4
product $4p$, $p$ prime, $p \equiv 1 \pmod 4$	5

Asymptotic facts and conjectures

• Higman-Sims asymptotic formula on number of groups of prime power order: This states that the number of groups of order $p^n$ is about $p^{(2n^3/27) + O(n^{8/3})}$.
• Conjecture that most finite groups are nilpotent
• Pyber's theorem on logarithmic quotient of number of nilpotent groups to number of groups approaching unity
• Higman's PORC conjecture states that the number of groups of order $p^n$ is a PORC function in $p$ for fixed $n$.

Properties

Supermultiplicativity

If $n = ab$ with $a$ and $b$ relatively prime, the number of groups of order $n$ is bounded from below by the product of the number of groups of orders $a$ and $b$ respectively. This is because we can take direct products for every pair of a group of order $a$ and a group of order $b$.