= Solution
For the <closed orientable surface> $\Sigma_g$, use the <fundamental group> presentation
$$
\pi_1(\Sigma_g)=\left\langle a_1,b_1,\ldots,a_g,b_g\mid\prod_{j=1}^g[a_j,b_j]=1\right\rangle.
$$
Since $g\ge1$, define a <homomorphism> to $\mathbb Z/n$ by sending $a_1$ to $1$ and every other displayed generator to zero. All commutators map to zero, so the relation is respected, and the map is surjective. Its kernel has index $n$. The <classification of connected covering spaces> supplies a connected $n$-sheeted <covering space>, including the identity cover when $n=1$.
Every connected finite <covering space> of $\Sigma_g$ is a <closed orientable surface>: <compactness> follows from finite degree, and the base orientation lifts through local covering charts. Lift a finite cell decomposition of the base. Every cell has $n$ lifts, so the <Euler characteristic under a finite covering> gives $\chi(\widetilde\Sigma)=n\chi(\Sigma_g)$. If the covering surface has genus $h$, this reads $2-2h=n(2-2g)$. Therefore
$$
\boxed{h=1+n(g-1).}
$$
This <genus of a finite cover of a closed orientable surface> depends only on the sheet number, not on the particular subgroup used to construct the cover.
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