A goal which I have been pursuing is to understand how number fields are distributed with respects to their invariants. To be more precise I was captivated by the following question: Let $N(X,n,G)$ be the number of number fields of dimension $n$ where $G$ is the Galois group of its Galois closure, and their discriminants is bounded by $|X|$ up to isomorphism. This question might be natural to ask: For which group $G\leq S_n$ one might get a positive proportion of number fields when $X\to \infty$.
From Class Field Theory or one can show it more elementary, using Delone-Faddeev correspondence, for $n=3$, $C_3$ has a density 0. And also when $G$ is an abelian group then the answer would be same as $C_3$ by using class field theory. Prof. Manjul Bhargava proved for $n=4$ , $D_4$ has a positive density, and also he showed for $n=5$ no other groups, except $S_5$, could have positive density.
I think, once Manjul told me, one might expect for $n=p$, $p$ is a prime number, the only group which can contribute with positive density is $S_p$. But I could not even find a heuristic that why this should be true.
Is there a heuristic or even a theorem which can support the above expectation? Or more generally what do we know about $N(X,n,G)$?