jmc

Let $s$ be the side length of the cube. The volume of the cube is $s^3$ and the sum of the lengths of the edges of the cube is $12s$. Therefore, we must have $s^3=6\cdot 12s$, so $s^3=72s$. Subtracting $72s$ from both sides gives $$s^3-72s=0,$$so $$s(s^2-72)=0,$$which means $$s=0\text{ or }s=\pm \sqrt{72}$$ Discarding the non-positive solutions, we find $s=\sqrt{72}=6\sqrt{2}$. The volume of the cube is $s^3=6^3\cdot (\sqrt{2}{)}^3=\boxed{432\sqrt{2}}$ cubic units.