A graphical characterization of the group Co 2 Hans Cuypers Department of Mathematics Eindhoven University of Technology Postbox 513, 5600 MB Eindhoven The Netherlands 18 September 1992 Abstract In this note we show that, up to isomorphism, there is a unique graph which is locally the distance 1–or–2 graph of the dual polar graph related to PSU 6 (2). The automorphism group of this graph is the second Conway group Co 2 . 1. Introduction It is well known, see for example [3,6,7,8], that the second Conway group Co 2 is the flag- transitive automorphism group of a geometry with Buekenhout diagram     c in which the residue of a point is the classical dual polar space related to PSU 6 (2). The point graph G of this geometry is a graph on 2300 points, such that for each vertex of this graph, the subgraph induced on the neighbors of the vertex, is the distance 1–or–2 graph of the dual polar graph related to PSU 6 (2). Denote by H this dual polar graph, and by H 1,2 its distance 1–or–2 graph. The group Co 2 acts as a rank 3 permutation group on the graph G , with point stabilizer P ΣU 6 (2). In this note we show that there is a unique graph which is locally H 1,2 . Theorem 1.1 Let Γ be a connected graph, such that for each vertex v of Γ, the induced subgraph Γ v on the neighbors of v is isomorphic to H 1,2 , then Γ is isomorphic to G . Meixner [6] and Yoshiara [8] have given characterizations of the geometry mentioned above, under the assumption that the automorphism group acts flag-transitively. 2. The dual polar graph related to PSU 6 (2) In this section we investigate the dual polar graph H related to the group PSU 6 (2). This graph is the point graph of a near hexagon with parameter set (s, t, t 2 ) = (2, 20, 4) and is distance regular with intersection array {42, 40, 32; 1, 5, 21}. We often identify the graph H with the near hexagon. 1