Maximal class group

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 $$p$$.

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

Weaker properties

 * UL-equivalent group