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 Frattini subgroup is always characteristic, so if it's Abelian, then it must be central, which is a contradiction of the orders.  So the Frattini subgroup is not Abelian:  Apply Lemma 4 from http://msp.org/pjm/1960/10-1/pjm-v10-n1-p12-s.pdf
