  We study the structure constants of the class algebra $R_Z(G_n)$ of the
wreath products $G_n$ associated to an arbitrary finite group G with respect to
the basis of conjugacy classes. We show that a suitable filtration on
$R_Z(G_n)$ gives rise to the graded ring $\mathcal G_G(n)$ with non-negative
integer structure constants independent of n (some of which are computed),
which are then encoded in a Farahat-Higman ring $\mathcal G_G$. The real
conjugacy classes of G come to play a distinguished role, and is treated in
detail in the case when G is a subgroup of $SL_2(C)$. The above results provide
new insight to the cohomology rings of Hilbert schemes of points on a
quasi-projective surface.
