On the universal embedding of the near hexagon related to the extended ternary Golay code

Let E 1 be the near hexagon on 729 points related to the extended ternary Golay code. We prove in an entirely geometric way that the generating and embedding ranks of E 1 are equal to 24. We also study the structure of the universal embedding e ˜ of E 1 . More precisely, we consider several nice subgeometries A of E 1 and determine which kind of embedding e ˜ A is, where e ˜ A is the embedding of A induced by e ˜ .