A note on the number of cyclic subgroups of a finite group

Let $G$ be a finite group, $L_1(G)$ be its poset of cyclic subgroups and consider the quantity $\alpha (G)=\frac{\mathrm{\mid }{L}_1(G)\mathrm{\mid }}{\mathrm{\mid }G\mathrm{\mid }}$. The aim of this paper is to study the class $\mathcal{C}$ of finite nilpotent groups having $\alpha (G)=\frac{3}{4}$. We show that if $G$ belongs to this class, then it is a 2-group satisfying certain conditions. Also, we study the appartenance of some classes of finite groups to $\mathcal{C}$.