The countable generic poset (P, ≤ ) is the Fraisse limit of the amalgamation class of finite partially ordered sets (see Glass et al., Math Z 214:55–66, 1993; Schmerl, Algebra Univers 9:317–321, 1979). It is homogeneous and \(\aleph_0\)-categorical with quantifier elimination. This paper concerns the structure (G, ∘ , ≤ ), where \((G,\circ)=\textrm{Aut}(P,\leq)\) and ≤ is the pointwise ordering on G. This is a natural structure to look at, because the ordering on G is ∅-definable up to reversal in the language { ∘ } (but this fact is not proved here). In this paper I show that (G, ≤ ) is elementarily equivalent to (P, ≤ ) itself. More generally, (G, ∘ , ≤ ) satisfies a weakening of the existential closure property for partially ordered groups. (Existential closure in groups has been studied for example in Higman and Scott.) This requires one to study the group G ∗ , obtained by freely adjoining a finite set of generators to G.