  The kissing number of $\mathbb{R}^n$ is the maximum number of
pairwise-nonoverlapping unit spheres that can simultaneously touch a central
unit sphere. Mittelmann and Vallentin (2010), based on the semidefinite
programming bound of Bachoc and Vallentin (2008), computed the best known upper
bounds for the kissing number for several values of $n \leq 23$. In this paper,
we exploit the symmetry present in the semidefinite programming bound to
provide improved upper bounds for $n = 9, \ldots, 23$.
