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 $p$.

Facts

For 2-groups

• Classification of finite 2-groups of maximal class: There are only three possibilities in general for any order -- the dihedral group, semidihedral group, and generalized quaternion group. For order 8, two of these collapse into one and we get only two possibilities.
• Finite non-abelian 2-group has maximal class iff its abelianization has order four: In particular, all the finite 2-groups of a given order and maximal class are isoclinic groups.

General facts

• P-group with derived subgroup of prime-square index not implies maximal class for odd p
• Group of exponent p and order greater than p^p is not embeddable in a maximal class group
• P-group is either absolutely regular or maximal class or has normal subgroup of exponent p and order p^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	$p^5$	$3+2\mathrm{g}\mathrm{c}\mathrm{d}(p-1,3)+\mathrm{g}\mathrm{c}\mathrm{d}(p-1,4)$

Relation with other properties

Weaker properties

• UL-equivalent group