Dempwolff group

Summary

In mathematical finite group theory, the Dempwolff group is a finite group of order 319979520 = 2¹⁵·3²·5·7·31, that is the unique nonsplit extension $2^{5\,.}\mathrm{GL}_{5}(\mathbb{F}_{2})$ of $\mathrm{GL}_{5}(\mathbb{F}_{2})$ by its natural module of order $2^5$. The uniqueness of such a nonsplit extension was shown by (Dempwolff 1972), and the existence by (Thompson 1976), who showed using some computer calculations of (Smith 1976) that the Dempwolff group is contained in the compact Lie group $E_{8}$ as the subgroup fixing a certain lattice in the Lie algebra of $E_{8}$, and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup.