Why is EXACT-CLIQUE not in co-NP?

Question

In my lecture I saw the problem of $\text{EXACT-CLIQUE} = \{\langle G,k\rangle : \text{the largest clique in $G$ is of order $k$}\}$

I understand this problem is obviously not in NP as we would need to check all cliques, but if we consider the complement of the problem, (i.e. is the largest clique not of order $k$), wouldn't this be in NP? Because we could simply use non-determinism to guess a clique bigger than $k$ and then check if it indeed makes a clique. My lecturer said however that the problem is not in co-NP but I did not understand the explanation.