What is the normal core of $GL_2(\mathbb{Z}_p)$ in $GL_2(\mathbb{Q}_p)$?

Question

What is the normal core of $GL_2(\mathbb{Z}_p)$ in $GL_2(\mathbb{Q}_p)$?

(the normal core of $H$ in $G$ is the largest subgroup of $H$ which is normal in $G$. it is the intersection of all $G$-conjugations of $H$)