Speaker recognition using support vector machines (SVMs) with features derived from generative models has been shown to perform well. Typically, a universal background model (UBM) is adapted to each utterance yielding a set of features that are used in an SVM. We consider the case where the UBM is a Gaussian mixture model (GMM), and maximum likelihood linear regression (MLLR) adaptation is used to adapt the means of the UBM. We examine two possible SVM feature expansions that arise in this context: the first, a GMM supervector is constructed by stacking the means of the adapted GMM, and the second consists of the elements of the MLLR transform. We examine several kernels associated with these expansions. We show that both expansions are equivalent given an appropriate choice of kernels. Experiments performed on the NIST SRE 2006 corpus clearly highlight that our choice of kernels, which are motivated by distance metrics between GMMs, outperform ad-hoc ones. We also apply SVM nuisance attribute projection (NAP) to the kernels as a form of channel compensation and show that, with a proper choice of kernel, we achieve results comparable to existing SVM based recognizers.