We give a presentation for the baseleaf preserving mapping class group $Mod(\mathcal{H})$ of the punctured solenoid $\mathcal{H}$. The generators for our presentation were introduced previously, and several relations among them were derived. In addition, we show that $Mod(\mathcal{H})$ has no non-trivial central elements. Our main tool is a new complex of triangulations of the disk upon which $Mod(\mathcal{H})$ acts.