# Unsupervised clustering in $10$ dimensions

I have a set of $\sim1000$ feature vectors in $\sim10$ dimensions and would like to cluster them in an unsupervised manner. I am expecting some of the vectors to bunch together in groups, but quite a lot to be outliers that are nowhere near each other (so $\sim5$ meaningful clusters and $1$ cluster which is just a uniform distribution in all dimensions).

I'm thinking of using a Gaussian mixture model; does that sounds reasonable? Is learning a GMM suitable for this higher dimension of data or is there perhaps a more suitable technique? Does $1000$ vectors sound like enough to do $10$-dimensional clustering. I am quite new to it so am trying to get a feel. Thanks very much for any insight you might be able to provide! :)