In this paper $\sigma$-permutable $n$-groups are defined and considered. An $n$-group $(G,f)$ is called $\sigma$-permutable, where $\sigma$ is a permutation of the set $\{1,\ldots,n+1\}$, iff $$ f(x_{\sigma 1}, łdots, x_{\sigma n}) = x_{\sigma (n + 1)} Łeftrightarrow f(x_1, łdots, x_n) = x_{n + 1} $$ for all $x_1,\ldots,x_{n+1}\in G$. Such $n$-groups are a special case of $\sigma$-permutable $n$-groupoids considered in [7] and also they represent a generalization of $i$-permutable $n$-groups from [6] and some other classes of $n$-groups. Examples of $\sigma$-permutable $n$-groups are given and some of their properties described. Necessary and sufficient conditions for an $n$-group to be $\sigma$-permutable are determined. Several conditions under which such $n$-groups are derived from a binary group are given.