On the intersection of conjugate subgroups of finite index

Famous Cayley theorem states that if a group $G$ possesses a subgroup $H$ of finite index $n$, then $G$ possesses a normal subgroup of index at most $n!$ and this subgroup can be obtained as the intersection of all conjugates of $H$. Considering a symmetric group of degree $n$ as $G$ and a point stabilizer (a symmetric group of degree $n-1$) as $H$ one can see, that the bound in this theorem is the best possible. It appeared that in many interesting case the bound on the index of the normal subgroup is much smaller, then $n!$, usually this bound is $n^c$, i.e., it is polynomial.