On a conjecture of Kahn for the Stiefel-Whitney classes of the regular representation (Q1355530)

Let \(G\) be a finite 2-group and let \(r_G\) denote the regular representation of \(G\). Define \(\nu(G)=\min\{n>0\mid w_{2^{n-1}}(r_G)\neq 0\}\) where \(w_j\) is the \(j^n\)th Stiefel-Whitney class. Let \(d(G)\) denote the minimal number of generators of \(G\). In [\textit{B. Kahn}, J. Algebra 144, No. 1, 214-247 (1991; Zbl 0777.20019)] it was conjectured that \(\nu(G)\geq d(G)\) and proved for \(d(G)\leq 3\). The case when \(d(G)=4\) was proved in [\textit{Pham Anh Minh}, J. Algebra 179, No. 2, 483-500 (1976; Zbl 0855.20043)]. In this paper the authors proves the conjecture when \(d(G)=5\).