A new flag-transitive affine plane of order bm$2^9$

Abstract

Finite flag-transitive affine planes have received much attention during the past fifty years because of their connections with other combinatorial objects such as spreads, planar functions, semifields and linearized polynomials. In 1964, Foulser completely determined the automorphism groups of finite flag-transitive affine planes. If a flag-transitive affine plane has a solvable automorphism group, then the affine plane is called solvable. The non-solvable flag-transitive affine planes have been completely classified in the 1990s. But the complete classification for the solvable case seems far out of reach. All known solvable flag-transitive affine planes can be classified into two types: $\\mathcal{C}$-planes and $\\mathcal{H}$-planes, where $\\mathcal{H}$-planes only occur in the odd characteristic case. In this paper, we construct the first flag-transitive affine plane of order $2^9$ over its kernel $\\bF_{2^3}$, which is not of type $\\mathcal{C}$ and the largest Singer subgroup of the translation complement has order $(2^3-1)(2^9+1)/9$.