There exist groups of prime power order having a maximal elementary abelian subgroup of prime-square order and normal rank equal to the prime for every prime

Statement
Suppose $$p$$ is a prime number. Then, there exists a group $$P$$ whose order is a power of $$p$$ having an elementary abelian subgroup of order $$p^2$$ not contained in any bigger elementary abelian subgroup (i.e., $$P$$ is a fact about::group of prime power order having a maximal elementary abelian subgroup of prime-square order, and such that the normal rank of $$P$$ is $$p$$: the largest possible size of an elementary abelian normal subgroup of $$P$$ is exactly $$p^p$$.

Related facts

 * Maximal elementary abelian subgroup of prime-square order implies normal rank at most the prime
 * Maximal elementary abelian subgroup of prime-square order implies rank at most the prime for odd prime
 * Maximal elementary abelian subgroup of order four implies subgroup rank at most four