Discrete degree of symmetry of manifolds

We define the discrete degree of symmetry $disc-sym(X)$ of a closed $n$-manifold $X$ as the biggest $m\ge 0$ such that $X$ supports an effective action of $(\mathbf{Z}\mathrm{/}r{)}^m$ for arbitrarily big values of $r$. We prove that if $X$ is connected then $disc-sym(X)\le 3n\mathrm{/}2$. We propose the question of whether for every closed connected $n$-manifold $X$ the inequality $disc-sym(X)\le n$ holds true, and whether the only closed connected $n$-manifold $X$ for which $disc-sym(X)=n$ is the torus $T^n$. We prove partial results providing evidence for an affirmative answer to this question.