Fundamental group after attaching a 2-cell

Attaching a disc to $X$ along a based loop $\alpha$ kills precisely the normal closure of its homotopy class:

A group presentation gives a two-dimensional cell complex with one vertex, one oriented loop for each generator, and one 2-cell attached along each relator. Its fundamental group is the presented group.

Attach three 2-cells to $S^1\vee S^1$ along loops representing $a^2$, $b^3$, and $(ab{)}^2$. The resulting group isIndeed, $a\mapsto (12)$ and $b\mapsto (123)$ gives a surjection to $S_3$, while $aba=b^{-1}$ reduces every word to one of $b^i$ or $a{b}^i$, so the presented group has at most six elements.