  Suppose $C$ is a cyclic Galois cover of the projective line branched at the
three points $0$, $1$, and $\infty$. Under a mild condition on the
ramification, we determine the structure of the graded Lie algebra of the lower
central series of the fundamental group of $C$ in terms of a basis which is
well-suited to studying the action of the absolute Galois group of
$\mathbb{Q}$.
