 Solution

The class of finite groups is not first-order axiomatizable. If $S$ axiomatized it, add constants $c_1,c_2,\dots$ and all inequalities $c_i\ne c_j$. Every finite fragment has a sufficiently large finite cyclic group as a model. The compactness theorem would then give an infinite group satisfying $S$, contradicting the definition of the class. This is the group instance of the fact that finite structures are not first-order axiomatizable.