Let $F$ be a field of order $p^a$ where $p$ is prime and $a\in \mathbb{N}$. For $n\geq2$ let $P\leq GL_n(F)$ be the group of upper unitriangular matrices, i.e. upper triangular matrices with all diagonal entries 1. What is the minimum size of a generating set for $P$ (rank of $P$) and what is the order of the Frattini subgroup of $P$?
My Attempt: I know that $P$ is a $p$-group of order $p^{n(n-1)/2}$. I tried to use the fact that $F$ is an $a$-dimensional vector space over its prime subfield $F_p$, so for $n=2$, I think we would have $rank(P)=a$ because given a basis $\{\alpha_1,...\alpha_a\}$ of $F$. It seemed to me that the set $\Bigg\{\begin{pmatrix} 1 & \alpha_i\\ 0 & 1 \end{pmatrix} \Bigg\}_{i=1}^{a}$ would be a minimal generating set for $P$. But it gets more complicated when $n\neq2$. Also I thought I would use Burnside's basis theorem to find the order of the Frattini subgroup of $P$ since it states that $rank(P)=rank(P/\phi(P))$ where $\phi(P)$ is the Frattini subgroup of $P$, but I am also not sure about this.
Any help will be appreciated. Thanks!