P-group of nilpotency class $2$ and Schur multiplier Let $p$ be an odd prime and let $G$ be a p-group of nilpotency class $2$ in which $V=G/G'$ is an elementary abelian p-group. Let $M(V)$ be the Schur multiplier of $V$ which is isomorphic to the exterior square $V\wedge V$.
Is $M(V)$ necessarily isomorphic to $G'$?
If the answer is no, does there exist a p-group $G$ of nilpotency class $2$ in which $V=G/G'$ is an elementary abelian p-group and $M(G)=1$?
Thank you in advance.
 A: The answer to the first question is “no”. Note that if $K=G\times C_p$, where $C_p$ is cyclic of order $p$, then $K’=G’$, and $K^{\rm ab}\cong G^{\rm ab}\times C_p$. In particular, if $G$ is of class two and $G^{\rm ab}$ is elementary abelian, then the same holds for $K$, but the rank of $K^{\rm ab}$ is strictly larger than that of $G$. That means that you can make $V\wedge V$ have arbitrarily large rank while not changing the commutator subgroup, so we cannot have that $M(V)$ is necessarily isomorphic to $G’$.
As for an example of a $p$ group $G$ of class two with $G^{\rm ab}$ elementary abelian and with trivial Schur multiplier, the easiest example is the extra-special $p$-group of order $p^3$ that is not of exponent $p$,
$$G = \langle x,y \mid x^{p^2}=y^p=[x,y,x]=[x,y,y]=1, x^p=[x,y]\rangle.$$
The same will hold for any group that is isoclinic to $G$.
The Schur multipliers of the four non-abelian groups of order $p^3$ are known classically; you can also read off that this group has trivial Schur multiplier from Theorem 49 in Certain homological functors of $2$-generator $p$-groups of class $2$ by A. Magidin and R.F. Morse, in Computational Group Theory and the Theory of Groups II, Contemporary Mathematics no. 511, American Mathematical Society, Providence RI, 2010, pp. 127-166. The group above corresponds to the tuple $(\alpha,\beta,\gamma;\rho,\sigma)=(1,1,1;0,1)$ in the notation of Theorem 1 of that paper.
