Let $\Gamma_i$ be the set of matrices in $GL_2(\mathbb{Z}_p)$ which are congruent to $1$ modulo $p^i$, that is they are the congruence subgroups. I know that $\Gamma_i$ is a pro-$p$ group and $\Gamma_i/\Gamma_{i+1}$ has cardinality $p^4$.
Is $\Gamma_{i}/Z(\Gamma_{i})$ again a pro-$p$ group?
Also, I would like to know the index of $\Gamma_{i+1}/Z(\Gamma_{i+1})$ inside $\Gamma_{i}/Z(\Gamma_{i})$ where $Z(\Gamma_{i+1})$ is the center of $\Gamma_{i+1}$.