Every group of prime power order is a subgroup of an iterated wreath product of groups of order p

Statement
Suppose $$G$$ is a fact about::group of prime power order, i.e., a finite $$p$$-group for some prime $$p$$. Then, $$G$$ can be embedded as a subgroup of a $$p$$-group $$P$$, where $$P$$ is an iterated wreath product of cyclic groups of order $$p$$. (Note that if $$P$$ is the wreath product of $$n$$ such groups, $$P$$ is isomorphic to the $$p$$-Sylow subgroup of the symmetric group on a set of size $$p^n$$).

Facts used

 * 1) uses::Cayley's theorem
 * 2) uses::Sylow implies order-dominating: The domination part of Sylow's theorem, which states that given a $$p$$-subgroup and a $$p$$-Sylow subgroup, some conjugate of the $$p$$-subgroup lies in the $$p$$-Sylow subgroup.