On the uniqueness of the generalized octagon of order (2,4)

The smallest known thick generalized octagon has order (2,4)(2,4) and can be constructed from the parabolic subgroups of the Ree group View the MathML sourceF42(2). It is not known whether this generalized octagon is unique up to isomorphism. We show that it is unique up to isomorphism among those having a point a whose stabilizer in the automorphism group both fixes setwise every line on a and contains a subgroup that is regular on the set of 1024 points at maximal distance to a. Our proof uses extensively the classification of the groups of order dividing 29. Keywords: Generalized polygon; Octagon; Point-line geometry; Incidence geometry; Groups of order a small power of 2; Ree group; Cayley graph