jmc

Let the three side lengths be $\frac{a}{r}$, $a$, and $ar$. Because the volume of the solid is $216{\text{ cm}}^3$,$$\frac{a}{r}\cdot a\cdot ar=216$$$$a=6$$The surface area of the solid is $288{\text{ cm}}^2$, so$$2(\frac{a^2}{r}+a^{2}r+a^2)=288$$Note that the sum of the side lengths of the cube is $4(\frac{6}{r}+6+6r)$ and that the equation above has a similar form.$$2(\frac{36}{r}+36r+36)=288$$$$2(\frac{6}{r}+6r+6)=48$$$$4(\frac{6}{r}+6r+6)=96$$The sum of all the edges of the cube is $\boxed{96}$ centimeters.