Generators in a $2$-generated $p$-group

Question

I suppose to have a 2-generated $p$-group $G$. I know that if $\langle a,b\rangle =G$, then $\langle aΦ(G), bΦ(G)\rangle = G/Φ(G)$, where $Φ(G)$ is the Frattini subgroup of $G$.

Is it also true that if I have $\langle cΦ(G),dΦ(G)\rangle = G/Φ(G)$, then $\langle c,d\rangle =G$ ?

Answer 1

Yes, this is (not just for 2, but any number) the statement of Burnside’s basis theorem.