Just want to add that $G$ has a quotient isomorphic to $H:=\mathbb Z_p \ltimes_\varphi \mathbb Z^{p-1}$ where $\varphi(1)\cdot v=Av$ for some $A \in SL_{p-1}(\mathbb Z)$ of order $p$; in particular, $G$ is infinite.
Indeed, such an $A$ exists: take $A$ to be a companion matrix to the polynomial $\frac{x^p-1}{x-1}$. Then any element in $H$ not contained in $ \mathbb Z_p {0} \times {0}$ \mathbb Z ^{p-1}$ has order $p$, so mapping $x$ to $(1,0)$ and $y$ to $(1,e_1)$ will give a well-defined homomorphism which can be shown to be surjective.
