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.