Thank you and my (prevciously) +1. BTW "Entire function of ln(x)" This remind me of the concept of "Dulac series" in the investigation of the finteness part of the Hilbert 16th problem

May be related to your question: mathoverflow.net/questions/46068/…

@mme I remember a similar phrase in Hirsch diff topology : ebery codimension one compact summanfiold of a compact manifold separate the space provided all thing are orientable

I dont know if this invariant is well defined and helpful or no?

@HJRW Thank you very much for your attention to my question. Yes, as you wrote, $n$ is allowed to depend on $G$. For finite group $G$ we have an effective action of $G$ on $\mathbb{R}^{|G|}$ since $G$ is embedded in $S_{|G|}$. So, according to the update version of the question, I think we may define the following invariant associated to group: The minimum of all $n$ for which there exists an effective action of $G$ on $\mathbb{R}^n$ such that the space of $G$ fixed vector is a one dimensional space. Some how similar to irreducible representation.

@YCor I think "Sym" is more popular since the notation $S_n$ used for symmetric group but every element of $S_n$ is called a permutation.

@Echo Ok in this sense only very huge groups does not admit injective morphisms $G\to Per(|mathbb{R}^n}$

@Echo thank you for your comment. But in the question we requiere that action is by isometry not exclusively by permutation. In the example we have permutation but in general we work with isometry. I think the answer of Prof. Valette completes the answer to this question since it explicitly represent the quadratic form