  The fundamental example of Gerstenhaber algebra is the space
$T_{poly}({\mathbb R}^d)$ of polyvector fields on $\mathbb{R}^d$, equipped with
the wedge product and the Schouten bracket. In this paper, we explicitely
describe what is the enveloping $G_\infty$ algebra of a Gerstenhaber algebra
$\mathcal{G}$. This structure gives us a definition of the Chevalley-Harrison
cohomology operator for $\mathcal{G}$. We finally show the nontriviality of a
Chevalley-Harrison cohomology group for a natural Gerstenhaber subalgebra in
$T_{poly}({\mathbb R}^d)$.
