Integral and BP cohomologies of extraspecial \(p\)-groups for odd primes (Q2386855)

Denote by \(p^5_+\) the extraspecial \(p\)-group of order \(p^5\) and exponent \(p\). Let \({G}\) be a central product of \(p^5_+\) by \(S^1\) or \({\mathbb Z}/p^m\). The author computes the integral cohomology, the Brown-Peterson cohomology and the Morava K-theory of \({G}\). In particular, he shows that \({\text{BP}}^{\text{ odd}}(Bp^5_+)=0\).