The authors prove that the only generalized hexagons of order (2,2) are the classical one and its dual (both associated with the group

${G}_{2}\left(2\right))$. They also prove the uniqueness of the generalized hexagon of order (2,8), associated with

${}^{3}{D}_{4}\left(2\right)$, and of the near octagon of order (2,4;0,3), associated with the Hall-Janko group. The treatment of the case (2,8) is facilitated by the use of results of Ronan and Timmesfeld. The arguments for all three cases are presented in terms of the (distance-regular) incidence graphs associated with the geometries. A central theme is the identification of subgraphs which are isomorphic to ”2-covers” of the n-cube.