No group of following property. Is this true? Let $p$ be a prime greater than 3 and $G$ be group of order $p^5$. Is it true that there is no group $G$ of order $p^5$ such that the order of frattini subgroup is $p^3$ and the order of center is $p^2$?
If the answer is yes, how to prove it.

The class $3$ quotient of the Burnside group $B(2,p)$ has these properties. It has the presentation
$\langle a,b,c,d,e \mid a^p=b^p=c^p=d^p=e^p=1, [b,a]=c, [c,a]=d, [c,b]=e, d,e\ {\rm central}\ \rangle$
The Frattini subgroup is $\langle c,d,e \rangle$ and the centre is $\langle d,e\rangle$.
