Group of prime power order having a maximal elementary abelian subgroup of prime-square order

Definition
Let $$P$$ be a group of prime power order, i.e., a finite $$p$$-group for some prime number $$p$$. We say that $$P$$ has a maximal elementary abelian subgroup of prime-square order if it satisfies the following equivalent conditions:


 * 1) There exists an elementary abelian subgroup $$A$$ of $$P$$ of order $$p^2$$ that is not contained in any bigger elementary abelian subgroup of $$P$$.
 * 2) There exists an elementary abelian subgroup $$A$$ of $$P$$ of order $$p^2$$ such that $$\Omega_1(C_P(A)) = A$$.
 * 3) There exists an element $$t \in P$$ such that $$\Omega_1(C_P(t))$$ is elementary abelian of order $$p^2$$.