Groups of prime power order

This is a specific information page. For the definition of the term and related information, see group of prime power order

This article gives information about groups of prime power order, i.e., order $p^n$ where $p$ is a prime number and $n$ is a nonnegative integer. The case $n = 0$, i.e., groups of order 1, only includes the trivial group, so we exclude this case from the discussion below.

Information

$n$	Discussion of groups of order $p^n$, generic $p$	Classification	Case $p = 2$	Case $p = 3$	Case $p =5$	Case $p = 7$	Case $p = 11$
generic	--	--	groups of order 2^n	groups of order 3^n	groups of order 5^n	groups of order 7^n	groups of order 11^n
1	only one group, see group of prime order	--	cyclic group:Z2	cyclic group:Z3	cyclic group:Z5	cyclic group:Z7	cyclic group:Z11
2	groups of prime-square order	See classification of groups of prime-square order. All groups of this order are abelian	groups of order 4	groups of order 9	groups of order 25	groups of order 49	groups of order 121
3	groups of prime-cube order	See classification of groups of prime-cube order	groups of order 8	groups of order 27	groups of order 125	groups of order 343	groups of order 1331
4	groups of prime-fourth order	See classification of groups of order 16, classification of groups of prime-fourth order for odd prime	groups of order 16	groups of order 81	groups of order 625	groups of order 2401	groups of order 14641
5	groups of prime-fifth order	?	groups of order 32	groups of order 243	groups of order 3125	groups of order 16807	groups of order 161051
6	groups of prime-sixth order	?	groups of order 64	groups of order 729	groups of order 15625	groups of order 117649	groups of order 1771561
7	groups of prime-seventh order	?	groups of order 128	groups of order 2187	groups of order 78125	groups of order 823543	groups of order 19487171
8	groups of prime-eighth order	?	groups of order 256
9	groups of prime-ninth order	?	groups of order 512
10	groups of prime-tenth order	?	groups of order 1024

Counts

$n$	Generic formula for number of groups of order $p^n$ (if it exists) for large enough $p$	Minimum $p$ from which the formula becomes valid	Number of groups of order $2^n$	Number of groups of order $3^n$	Number of groups of order $5^n$	Number of groups of order $7^n$	Number of groups of order $11^n$
1	1	2	1	1	1	1	1
2	2	2	2	2	2	2	2
3	5	2	5	5	5	5	5
4	15	3	14	15	15	15	15
5	$2p + 61 + 2\operatorname{gcd}(p-1,3) + \operatorname{gcd}(p-1,4)$	5	51	67	77	83	87
6	$3p^2 + 39p + 344 + 24 \operatorname{gcd}(p - 1,3) + 11 \operatorname{gcd}(p-1,4) + 2 \operatorname{gcd}(p-1,5)$	5	267	504	684	860	1192
7	$3p^5 + 12p^4 + 44p^3 + 170p^2 + 707p + 2455 + (4p^2 + 44p + 291) \operatorname{gcd}(p-1, 3)$ $+ (p^2 + 19p + 135) \operatorname{gcd}(p-1, 4)$ $+ (3p + 31) \operatorname{gcd}(p-1, 5) + 4 \operatorname{gcd}(p-1, 7) +5 \operatorname{gcd}(p-1, 8)$$+ \operatorname{gcd}(p-1, 9)$	7	2328	9310	34297	113147	750735
8	unknown		56092	1396077	unknown	unknown	unknown
9	unknown		10494213	unknown	unknown	unknown	unknown
10	unknown		49487365422	unknown	unknown	unknown	unknown