On the 2-compact group DI(4)

Besides the simple connected compact Lie groups there exists one further simple connected 2-compact group, constructed by Dwyer and Wilkerson, the group DI(4). The mod-2 cohomology of the associated classifying space BDI(4) realizes the rank 4 mod-2 Dickson invariants. We show that mod-2 cohomology determines the homotopy type of the space BDI(4) and that the maximal torus normalizer determines the isomorphism type of DI(4) as 2-compact group. We also calculate the set of homotopy classes of self maps of BDI(4).