Enumerating Galois extensions of number fields

Let k be a number field.  We provide an asymptotic formula for the number of Galois extensions of k with absolute discriminant bounded by some X, as X tends to infinity.  We also provide an asymptotic formula for the closely related count of extensions of k whose normal closure has discriminant bounded by X.  The key behind these results is a new upper bound on the number of Galois extensions of k with a given Galois group G.  This new upper bound improves over the previous best bound due to Ellenberg and Venkatesh, and is the first bound for general G with an exponent that decays as the order of G tends to infinity.