Solution:By Orbit-Stabilizer theorem,$\vert G:G_g\vert$.$\vert G_g:Z(G) \vert$=$\vert G:Z(G)\vert =n \implies \vert G:G_g\vert$ divides $n$.
When the action is by conjugation,the no. of conjugates of $g$ is $\vert G:C_G(g)\vert $.
Thus,$\vert G:C_G(g)\vert $ divides $n$,which implies that every conjugacy class has atmost $n$ elements.
I'm not sure whether this is correct,please tell me if i'm missing some detail.
Also,i don't know how to prove this in another way.
Please help in this.