Large orbits of nilpotent subgroups of linear groups

Suppose that $G$ is a finite solvable group and $V$ is a finite, faithful and completely reducible $G$-module. Let $N$ be a nilpotent subgroup of $G$, then there exits $v\in V$ such that $\mathrm{\mid }{\text{bC}}_N(v)\mathrm{\mid }\le (\mathrm{\mid }N\mathrm{\mid }\mathrm{/}p{)}^{1\mathrm{/}p}$, where $p$ is the smallest prime divisor of $\mathrm{\mid }N\mathrm{\mid }$.