It is difficult to come up with a consistent notation for all groups of a certain order since their construction is somewhat chaotic. We might be able to describe all the groups of order $p^3$ or $p^4$ but what about all groups of order $p^6$? Or order $p^4q^2$?
The software package GAP (http://www.gap-system.org/) has a catalogue of all groups of order up to 2000 or so and so I've sometimes referred to groups by their catalogue number, for example, SmallGroup(96, 33) refers to a particular group in that library. (As does SmallGroup(512, 1000000)!)