Classifying all groups of order $2012$? Let $G$ be a group such that $|G| = 2012$, how would you classify, up to isomorphism, all groups $G$? 
Clearly $2012 = 503 \times 2 \times 2$ and so $G \cong C_{503} \times C_2 \times C_2$ but how would you find the others?
 A: First: there are $\,2\,$ non-isomorphic abelian groups of order $\,2012\,$, which have already been mentioned by you and Chris.
Second: as Jyrki mentioned, if $\,|G|=2012\,$ then $\,G\,$ always has a normal sbgp. $\,P\,$ of order $\,503\,$, so $\,G\,$ is always an extension of such a sbgp. Since $\,|\operatorname{Aut}(P)|=502=2\cdot 251\,$ , we have at least two possible such extensions. Putting $\,P:=\langle p\rangle\,\,,\,C_4:=\langle c\rangle\,\,,\,C_2\times C_2:=\langle a,b\rangle$:$$(i) P\rtimes C_4\,\,,\,\text{with homomorphism}\,\,\,C_4\to\operatorname{Aut}(P)\,\,\,\text{defined by}\,\,\,p^c:=p^{-1}$$
$$(ii)P\rtimes\left(C_2\times C_2\right)\,\,,\,\text{with hom.}\,\, C_2\times C_2\to\operatorname{Aut}(P)\,\text{defined by}\,\,p^a:=p^{-1}\,,\,p^b:=p$$
The other "obvious" action of $\,C_2\times C_2\,$ on $\,P\,\,:p^b=p^{-1}\,,\,p^a=p\,$ gives us a semidirect product isomorphic with (ii) above, as we've an automorphism of the Klein group mapping each generator into the other one.
Thus, we've $\,4\,$ non-isomorphic groups of order $\,2012$
