Is this 2D structure triclinic? 
The only rotation axis obvious to me is rotation by 360 degrees, the identity. Vertical mirror planes I've been dicing and cutting it through several planes and I still see none. Yet, the structure looks fairly symmetric and I feel like I'm missing something.
No center of inversion either.
Not too terribly sure about horizontal mirror planes. I suppose that one bisecting the plane horizontally would be there but I am not entirely sure if this is appropriate.
Right now I'm saying this structure belongs to the C1 point group.
 A: I have consulted the International Tables for Crystallography (which is the authorative reference for symmetries, point groups, space groups and the like; unfortunately, it is not freely available on the web), and as drawn in the question, so the plane group is indeed p1, so no rotation axes.
However, it would be nice if you could please clarify how you have specified the basis: When you say it is $\{2/3, 1/3\}$, does that mean $2\hat{\mathbf{x}}/3 + \hat{\mathbf{y}}/3$ or $2\mathbf{a}/3 + \mathbf{b}/3$? The latter is the most frequently used method of specifying a basis, and the international tables actually list the coordinates $2\mathbf{a}/3, \mathbf{b}/3$ as a position that is compatible with a threefold rotation, and hence plane group p3m1. Could it be that you have misunderstood the problem formulation?
I'll add a technical note add the end: In two dimensions, the p1 plane group has the oblique lattice. Triclinic is a beast belonging to three dimensions (and one of its space groups is P1, note the capital P).
