  Let $(S,H)$ be a generalized Fermat pair of the type $(k,n)$. If $F\subset S$
is the set of fixed points of the non-trivial elements of the group $H$, then
$F$ is exactly the set of hyperosculating points of the standard embedding
$S\hookrightarrow {\mathbb{P}}^{n}$. We provide an optimal lower bound (this
being sharp in a dense open set of the moduli space of the generalized Fermat
curves) for the Weierstrass weight of these points.
