  We classify tensors with maximal and next to maximal dimensional symmetry
groups under a natural genericity assumption (1-genericity), in dimensions
greater than 7. In other words, we classify minimal dimensional orbits in the
space of (m,m,m) tensors assuming 1-genericity. Our study uncovers new tensors
with striking geometry. This paper was motivated by Strassen's laser method for
bounding the exponent of matrix multiplication. The best known tensor for the
laser method is the large Coppersmith-Winograd tensor, and our study began with
the observation that it has a large symmetry group, of dimension m^2/2 +m/2. We
show that in odd dimensions, this is the largest possible for a 1-generic
tensor, but in even dimensions we exhibit a tensor with a larger dimensional
symmetry group. In the course of the proof, we classify nondegenerate bilinear
forms with large dimensional stabilizers, which may be of interest in its own
right.
