  We resolve most cases of identifiability from sixth-order moments for
Gaussian mixtures on spaces of large dimensions. Our results imply that the
parameters of a generic mixture of $ m\leq\mathcal{O}(n^4) $ Gaussians on $
\mathbb R^n $ can be uniquely recovered from the mixture moments of degree 6.
The constant hidden in the $ \mathcal{O} $-notation is optimal and equals the
one in the upper bound from counting parameters. We give an argument that
degree-4 moments never suffice in any nontrivial case, and we conduct some
numerical experiments indicating that degree 5 is minimal for identifiability.
