So I know that, for a stabilizer code, the stabilizer group $S$ has $n-k$ commuting generators.
Is there a general way of knowing what the order of the full group of $S$ is, aside from writing out all the possible combinations of its generators $g_{1}, \dots, g_{n-k}$ which give rise to different/unique stabilizers?
My intuition says no, because $|S|$ is dependent on the stabilizer generators, which are different for each code. For example, for one code $g_{1}g_{2}=g_{3}g_{4}$, and so, we cannot count this stabilizer twice in $S$. However, for another code $g_{1}g_{2} \neq g_{3}g_{4}$, giving two new stabilizers to $S$. However, I would be grateful if somebody could confirm/debunk my suspicions.