Let $x$ be a fixed vector in the carrier vector space of an irrep $\rho$ of a compact Lie group $G$, and let $G_x$ be the stabilizer subgroup of $G$ with respect to $x$. Assume that $x$ is not invariant under infinitesimal transformations, i.e. $G_x$ is discrete. How can I compute the size of $G_x$? A more ambitious question: how I can find all elements of $G_x$?

My interest is not abstract but computational. In practice I have $G=O(N)$ and $\rho$ is the symmetric traceless tensor representation of rank 4. The vector $x$ is given numerically, with high precision (e.g. 200 digits), so what I am looking for is a numerical procedure. Formally I could extract $|G_x|$ from a group integral over $O(N)$ with the Haar measure: $$ \int dg\, \delta(x- x^g) $$ What would be the best way to implement this in practice? One way would be to approximate the delta-function by a peaked gaussian, and brute force the integral via Monte Carlo integration. Is there is a smarter way to procede?