The class of nilpotent groups of class $c$ and prime exponent $p\,{>}\,c$ with additional predicates $P_c\,{\subseteq}\,P_{c-1}\,{\subseteq}\,\ldots\,{\subseteq}\,P_1$ for suitable subgroups has the amalgamation property. Hence the Fraïssé limit $D$ of the finite groups of this class exists. $\langle 1\rangle\,{\subseteq}\,P_c(D)\,{\subseteq}\,\ldots\,{\subseteq}\,P_2(D)\,{\subseteq}\,P_1(D)\,{=}\,D$ is the lower and the upper central series of $D$. In this extended language, $D$ is ultrahomogeneous. The elementary theory of $D$ allows the elimination of quantifiers and it is $\aleph_0$-categorical. For $c\,{=}\,2$ this was proved by Baudisch in Bull. London Math. Soc. 33 (2001) 513–519.