Functor from a group to a poset Are there examples of functors from the category of a single group to the category a partially ordered set (some sort of representation of the group in a poset) ?
 A: Maybe the question (and the elided second question) wants to be rewritten as asking about posets of subgroups of a finite group, and asking about construction of the "incidence geometry", a simplicial complex attached to that poset?
I suggest this revision because it has at least one worthwhile answer/discussion, namely, for Coxeter groups ... definable as groups $G$ generated by sets $S$ of reflections (elements $s\in S$ satisfy $s^2=e$), defined by relations $(st)^{m_{s,t}}=e$ for $s,t\in S$ for some integer $m_{s,t}$ (by convention $m_{s,t}=\infty$ if no power of $st$ is $e$). The simplest example is the symmetric group on $n$ things, which is generated by "reflections" consisting of "adjacent" transpositions. 
The special subgroups of $G$ are those generated by subsets of the set of reflections. Two special subgroups are incident when their intersection is also special. Collections of mutually incident special subgroups are taken as simplices (maybe inclusion-reversing), and we obtain a simplicial complex (the associated "Coxeter complex").
It is a not-completely-trivial result that the resulting complex is a triangulation of a sphere for finite Coxeter groups. For certain infinite Coxeter groups ("affine" ones) the Coxeter complex is a triangulation of an affine space.
Another, related case of a meaningful simplicial complex attached to a group is the (spherical) building attached to parabolic subgroups $P$ of a reductive group $G$: two parabolics are incident when their intersection is another parabolic, and mutually incident collections are simplices (inclusion-reversing). These buildings are constructible by sticking together the Coxeter complexes ("apartments") made from conjugates of "Weyl groups". For example, for $G=GL_n(k)$ with a field $k$, the "Weyl group" $W$ is identifiable with permutation matrices in $G$, so is a Coxeter group, and the "apartments" are spheres.
More of this sort of discussion is possible, in many directions, ...
Edit: given the poser's (re-) framing of the question in the context of transformational (mathematical) music theory (an interesting idea, about which I read a few things some years ago...): very likely, the operationally useful idea would be that of "a group acting on [something]". Keyphrase "acting on", rather than "group qua group". Yes, relevant posets can easily arise, as "subsequent" states/ideas in a sequence of such, with the ordering simply being (essentially) "later", in a piece of music that is deterministically performed. (!) Even more literally, since we do not imagine music going backward in time, the truly relevant notion is semi-group acting on poset. [But now I must stop, because I have not thought about actual applications to music beyond this mild deconstruction of modern "theoretical music theory", and, also, I'm more sympathetic to the "tonal" tradition, and do not see music as essentially mathematical, any more than I see playing outfield as such... and, by this, I don't mean to disparage anyone...]
