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

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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 $p^{2}$.