  In this article, we prove that the full automorphism group of the
Z_2-orbifold of the Barnes-Wall lattice vertex operator algebra of central
charge 32 has the shape $2^{27}.E_6(2)$. In order to identify the group
structure, we introduce a graph structure on the Griess algebra and show that
it is a rank 3 graph associated to $E_6(2)$.
