On the Number of Planes in Neumaierʹs A8-Geometry

In one of his papers [2], A. Neumaier constructed a rank 4 incidence geometry on which the alternating group of degree 8 acts flag-transitively. This geometry is quite important since its point residue is the famous A7-geometry which is known to be the only flag-transitive locally classical C3-geometry which is not a polar space (see [1]). By counting chambers, we prove that the A8-geometry has 70 planes. This can be found in a paper of Pasiniʹs [4] without proof, but Neumaierʹs original paper only mentions 35 planes.