  Let $n$ be a positive integer and $G$ be a transitive permutation subgroup of
$S_n$. Given a number field $K$ with $[K:\mathbb{Q}]=n$, we let $\widetilde{K}$
be its Galois closure over $\mathbb{Q}$ and refer to
$Gal(\widetilde{K}/\mathbb{Q})$ as its Galois group. We may identify this
Galois group with a transitive subgroup of $S_n$. Given a real number $X>0$, we
set $N_{n}(X;G)$ to be the number of such number fields $K$ for which the
absolute discriminant is bounded above by $X$, and for which
$Gal(\widetilde{K}/\mathbb{Q})$ is isomorphic to $G$ as a permutation subgroup
of $S_n$. We prove an asymptotic upper bound for $N_n(X;G)$ as
$X\rightarrow\infty$. This result is conditional and based upon the
non-vanishing of certain polynomial determinants in $n$-variables. We expect
that these determinants are non-vanishing for many groups, and demonstrate
through some examples how they may be computed.
