Order of nilpotency of ideal generated by kernel of induced map between cohomology rings of 1- dimensional CW complexes

Question

Let $G$ be a $1$-dimensional CW complex and let $\Delta^*: H^*(G \times G) \to H^*(G)$ be the homomorphism induced by the diagonal map $\Delta : G \to G \times G$.

How do you compute the order of nilpotency of the ideal Ker$\Delta^*$ ?