= Solution
Every nonidentity <biholomorphism> in the finite <group> $G$ has only finitely many fixed points, by the <identity theorem on a Riemann surface> and <compactness>. Because the <group action> is a <faithful group action>, the union of these finite sets does not exhaust $X$. A point outside this union has trivial <stabilizer subgroup>, so its <group orbit> has $|G|$ points by the <orbit-stabilizer theorem>. The generic <fiber of a function> of the projection is such an <group orbit>, giving
$$
\boxed{\deg f=|G|.}
$$
More precisely, put $H=G_x$ and $m=|H|$. Differentiation at $x$ gives a <group homomorphism> $H\to\mathbb C^\times$. It is <injective>: if a nonidentity finite-order germ had the form $h(z)=z+a_kz^k+O(z^{k+1})$ with $k\geq2$ and $a_k\ne0$, then its $q$th iterate would begin $z+qa_kz^k$, so no positive iterate could be the identity. A germ equal to the identity gives the identity on $X$ by the <identity theorem on a Riemann surface>. Thus $H$ is a <cyclic group>, with derivative image the $m$th <roots of unity>.
For completeness, an explicit local coordinate linearizing $H$ is
$$
w(p)=\frac1m\sum_{h\in H}h'(x)^{-1}z(hp).
$$
It has $w'(x)=1$, and reindexing the sum gives $w(hp)=h'(x)w(p)$. Therefore a coordinate on the local <quotient space> is $w^m$. Other translates of a sufficiently small <neighbourhood> of $x$ are disjoint unless their translating element lies in $H$, so this local <quotient space> is also a chart for $X/G$. The <local cyclic quotient of a Riemann surface> consequently gives
$$
\boxed{\nu_f(x)=|G_x|.}
$$
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