homomorphism inducing Galois cover

Question

We are given a homomorphism $\rho: \pi_1(\Sigma_g - \{p_1,...,p_k\}) \rightarrow G$ , where $\Sigma_g$ is genus g Riemann surface and $p_i$'s are points on it, G is a finite group. Then it is claimed that "$\rho$ induces a Galois cover $p$ of $\Sigma_g$" s.t. $p: X \rightarrow \Sigma_g$ , $X$ is a Riemann surface. Does it mean that the Galois group of the cover is G, degree of the cover is |G|, and $p_i$'s are branch points? If so, what is the good reference to learn this? If not, what does this mean? Thanks in advance.