Let us denote by $G:= GL_2(\mathbb{Q}_p), G_0:= GL_2(\mathbb{Z}_p), g:= \begin{bmatrix}0 & 1\\p& 0\end{bmatrix}$ and by $G_1:= g G_0 g^{-1}$. I want to know if we have a good description of the subgroup generated inside $GL_2(Q_p)$ by union of $G_0$ and $G_1$, i.e., $<G_0\cup G_1>\subseteq G$. First of all by restriction on determinants we know any matrix in this subgroup will have determinant in $\mathbb{Z}_p^{*}$. Is that the only restriction, that is, can we get any matrices inside $G$ with determinant in $\mathbb{Z}_p^{*}$?
More generally if we have $K_1:= hG_0h^{-1}$ and $K_2:= hG_1h^{-1}$ with $h \in GL_2(\mathbb{Q}_p)$ then what can we say about $<K_1\cup K_2>$?
I have done some computations which shows I can make p-adic valuation of certain entries of the matrices in this group arbitrarily small, but not entirely sure how to proceed concretely from there or if this is a reasonable question to ask.
Thanks in advance.