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