On the number of planes in Neumaier's A(8)-geometry

In one of his papers [2], A. Neumaier constructed a rank 4 incidence geomet ry on which the alternating group of degree 8 acts flag-transitively. This geometry is quite important since its point residue is the famous. A(7)-geo metry which is known to be the only flag-transitive locally classical C-3-g eometry which is not a polar space (sce [1]). By counting chambers, we prov e that the A(8)-geometry has 70 planes. This can be found in a paper of Pas ini's [4] without proof, but Neumaier's original paper only mentions 35 pla nes. (C) 2001 Academic Press.