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		49487367289	unknown	unknown	unknown	unknown