In this paper we prove a generalization of the Hosszú--Gluskin theorem for $(sm,m)$-groups in terms of a $\{1,(s-1)m+1\}$-neutral operation and we define the algebra $(Q^m,\{\cdot,\varphi,c^m_1\})$ associated to the $(sm,m)$-group $(Q,A)$. The central operation of an $(n,m)$-group is defined in [7]. Research results of central operation properties using a bijection $\sigma_\alpha:Q^m\to Q^m$ are presented by means of a series of theorems. Then, a central operation of an $(sm,m)$-group is investigated using the previously mentioned algebra.