Let \(H\) be a finite group, let \(\mathbf V\) be the variety generated by \(H\) and for \(r\geq 1\) let \(G_r\) be the relatively free group in \(\mathbf V\) on \(r\) free generators. Results of \textit{G. Birkhoff} [Proc. Camb. Philos. Soc. 31, 433-454 (1935; Zbl 0013.00105)] and \textit{B. H. Neumann} [Math. Ann. 114, 506-525 (1937; Zbl 0016.35102)] show that \(G_r\) embeds as a subgroup of the direct product of \(|H|^r\) copies of \(H\) (although as Neumann points out this bound can be reduced substantially in some cases). Accordingly the authors let \(t(r)\) be the smallest positive integer \(t\) such that \(G_r\) can be embedded as a subgroup of the direct product of \(t\) copies of \(H\). Then \(t(r)\leq|H|^r\) and the paper is concerned with finding \(t(r)\) for various types of group \(H\). (The function \(t(r)\) is also dependent upon \(H\) of course.) The authors determine the structure of \(G_r\) and the exact values of \(t(r)\) in the cases when \(H\) is a minimal non-Abelian finite group or a dihedral group \(D_n\) with \(n\) odd. The latter work generalizes a result of \textit{B. Fine} [Arch. Math. 46, 193-197 (1986; Zbl 0569.20021)]. The authors also obtain \(t(r)\) when \(H\) is a minimal non-nilpotent group and they show that a group \(H\) is nilpotent of class at most \(c\) if and only if \(t(r)\) is bounded by a polynomial in \(r\) of degree at most \(c\).