Characterizing finite $p$-groups by their Schur multipliers, $t(G)=5$

Let $G$ be a finite $p$-group of order $p^n$. It is known that $\mathrm{\mid }\mathcal{M}(G)\mathrm{\mid }=p^{\text{f}12n(n-1)-t(G)}$ and $t(G)\ge 0$. The structure of $G$ characterized when $t(G)\le 4$ in \cite{be,el,ni,sa,zh}. The structure description of $G$ is determined in this paper for $t(G)=5$.