On Davenport Constant of the Group $C_2^{r-1} \oplus C_{2k}$

Let $G$ be a finite abelian group. The Davenport constant $\mathsf{D}(G)$ is the maximal length of minimal zero-sum sequences over $G$. For groups form $C_2^{r-1} \oplus C_{2k}$ known for $r\leq 5$. In this paper, we get precise value $\mathsf{D}(C_2^{5} C_{2k})$ $k\geq 149$. It also worth pointing out that our result can imply $\mathsf{D}(C_2^{4} C_{2k})$.