  We prove that a locally nilpotent group $G$ of $C^{1}$ diffeomorphisms of a
compact surface $S$ of non-vanishing Euler characteristic has a finite orbit
${\mathcal O}$ whose cardinal is bounded by above by a function of the
characteristic of Euler of $S$.
  We focus on the case of negative Euler characteristic $\chi (S)$. Then we can
choose ${\mathcal O}$ so that it consists of global contractible fixed points
of thesubgroup $G_0$ of $G$ consisting of isotopic to the identity elements. In
particular $G$ has a global contractible fixed point if it consists of isotopic
to the identity elements.
