# Identifying the numbers of degree $n$ covering spaces of $X$

Let $X$ be a path-connected, locally path-connected and semilocally simply-connected space. Can we find a correspondence between degree $n$ covering spaces of $X$ and group homomorphism $\pi_1(X)\rightarrow S_n$? ($S_n$ is the permutation group)

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From the classification of covering spaces of such a space, we know they are in correspondence with the subgroups of the fundamental group. Can you relate the index of the subgroup with some useful parameter of the covering? –  Mariano Suárez-Alvarez Aug 21 '12 at 5:33
Don't the subgroups of index n correspond to connected coverings only? –  mland Aug 21 '12 at 7:56
Connected $n$ covering spaces are in correspondence with the orbits of all index $n$ subgroup acted by conjugation. I cannot find an easy way to identify these orbits. –  Hezudao Aug 21 '12 at 14:38