# Maximal class group

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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## Definition

A maximal class group is a group of prime power order that has nilpotency class $c$, and order $p^{c+1}$, where $p$ is a prime number and $c > 1$. Equivalently, it has the following properties:

1. The abelianization of the group, i.e., the quotient of the group by its commutator subgroup has order $p^2$.
2. The upper central series and lower central series coincide, and all the successive quotients (except the top-most quotient) are of order $p$.

## Examples

Prime number $p$ Nilpotency class $c$ $p^{c+1}$ Number of maximal class groups List of maximal class groups List of GAP IDs second part (in order of list)
2 2 8 2 dihedral group:D8, quaternion group 3,4
2 3 16 3 dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16 7,8,9
2 4 32 3 dihedral group:D32, semidihedral group:SD32, generalized quaternion group:Q32 18,19,20
2 5 64 3 dihedral group:D64, semidihedral group:SD64, generalized quaternion group:Q64 52,53,54
3 2 27 2 prime-cube order group:U(3,3), M27 3,4
3 3 81 4 wreath product of Z3 and Z3, SmallGroup(81,8), SmallGroup(81,9), SmallGroup(81,10) 7,8,9,10
3 4 243 6
odd $p$ 2 $p^3$ 2 prime-cube order group:U(3,p), semidirect product of cyclic group of prime-square order and cyclic group of prime order 3,4
odd $p$ 3 $p^4$ 4 7,8,9,10
odd $p$ 4 $p^5$ $3 + 2\operatorname{gcd}(p-1,3) + \operatorname{gcd}(p-1,4)$