Central series quotient of wreath product of groups of order p

Definition
A central series quotient of wreath product of groups of order p is a quotient of a wreath product of groups of order p by a member of its lower central series (which in this case equals the upper central series, because the group is a maximal class group).

Particular cases

 * 1) The abelianization of a wreath product of groups of order $$p$$ is an elementary abelian group of prime-square order, i.e., a direct product of two copies of the cyclic group of order $$p$$.
 * 2) The quotient of the group by its commutator with its commutator subgroup is a non-abelian group of order $$p^3$$. For odd $$p$$, it is isomorphic to prime-cube order group:U3p, the unique non-abelian $$p$$-group of order $$p^3$$ and exponent $$p$$. For $$p = 2$$, it is isomorphic to dihedral group:D8.
 * 3) The inner automorphism group of wreath product of groups of order p is a group of order $$p^p$$. It is a regular p-group. For odd $$p$$, it is also a maximal class group of exponent $$p$$, and is not an absolutely regular p-group.

Note that for $$p = 2$$, (1) and (3) coincide, and (2) coincides with the whole group. For $$p = 3$$ (2) and (3) coincide.