# Are there Galois covers of curves branched at 1 point?

Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one point?

I know that this cannot happen when $G$ is abelian, but what about the general case?

(EDIT: a branch point is the image in $C/G$ of a ramification point)

• Interesting question! Why can't it happen if $G$ is abelian? – Bruno Joyal Nov 22 '13 at 5:35
• Yes it can if $G$ is a center-free group. – syxiao Dec 10 '18 at 1:08