Group of exponent p and order greater than p^p is not embeddable in a maximal class group

Statement in terms of non-embeddability
Suppose $$p$$ is a prime number and $$P$$ is a finite $$p$$-group (i.e., $$P$$ is a group of prime power order with $$p$$ the prime) such that $$P$$ is a group of exponent p and the order of $$P$$ is at least $$p^{p+1}$$. Then, there is no fact about::maximal class group containing $$P$$.

Statement in terms of absence of subgroups
Suppose $$p$$ is a prime number and $$G$$ is a finite $$p$$-group that is a maximal class group. Then, $$G$$ does not contain any subgroup of exponent p and order $$p^{p+1}$$.

Caveat
Note that it is possible for a maximal class group to contain subgroups that are generated by elements of order $$p$$. Thus, it is perfectly possible to have a maximal class group $$G$$ of order greater than $$p^p$$ such that $$\Omega_1(G) = G$$. The standard example is a wreath product of groups of order p, which is a maximal class group of order $$p^{p+1}$$ generated by elements of order $$p$$ but has order $$p^2$$. (It is, in fact, isomorphic to the $$p$$-Sylow subgroup of the symmetric group of degree $$p^2$$).