I want to ask a question on character theory of finite group.

Let $G$ be a finite nonablian group， and let $p$ be the minimal divisior of |G|=n. Suppose $\chi\in IRR(G)$. Let $A=Gal(Q_n/Q)$. $A$ acts on $IRR(G)$ as usual. My question is: if $\chi$ vanishes on all conjugacy classes but $p$ conjugacy classes, is the length of the $A$-orbit with $\chi$ $p-1$？

Thank you in advance.