This is a past exam question (so no solutions are provided) and it asks to prove that if a group $G$ with $27$ elements acts on a set $X$ with $135$ elements then for every $x_i$, $x_j$ in $X$ the stabilizers coincide; $Stab(x_i)= Stab(x_j)$.
What I did was to write the class equation $$135=|X|=\sum_{x_i}|O(x_i)|$$ where the $x_i$s are orbit representatives ($ O(x_i)$ denotes the orbit of $x_i$). Then I substituted $|O(x_i)|$ with $|G|/|Stab(x_i)|$ and I am here $$27\cdot 5=135= \sum_{x_i}\frac{|G|}{|Stab(x_i)|}$$ which in turn gives $$5= \sum_{x_i}\frac{1}{|Stab(x_i)|}$$ but I don't really know what to do know. I've solved many exercises with actions that were rather trivial but this one seems to need something else... Thanks in advance!