Computing the stabilizer of a specific vector in a Lie group representation

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?