This Note gives an extension of Mahler's theorem on lattices in ${\mathbb{R}}^n$ to simply connected nilpotent groups with a $Q$-structure. From this one gets an application to groups of Heisenberg type and a generalization of Hermite's inequality.

We show that the unimodular lattice associated to the rank 20 quaternionic matrix group ${\mathbf{SL}}_2\left({\mathbf{F}}_{41}\right)\otimes {\tilde{S}}_3\subset {\mathbf{GL}}_{80}\left(\mathbf{Z}\right)$ is a fourth example of an 80-dimensional extremal lattice. Our method is to use the positivity of the $\Theta$-series in conjunction with an enumeration of all the norm 10 vectors. The use of Aschbacher’s theorem on subgroups of finite classical groups (reliant on the classification of finite simple groups) provides one proof that this lattice is distinct from the previous three, while computing the inner product...