The volume of a cube is given by the formula:

$V = s^3$

where $s$ is the side length of the cube. Given that the volume is $512 \, \text{cm}^3$, we can find $s$ by taking the cube root of the volume:

$s = \sqrt[3]{512} = 8 \, \text{cm}$

Now, to find the total surface area $A$ of the cube, we use the formula:

$A = 6s^2$

Substituting $s = 8 \, \text{cm}$:

$A = 6 \times (8)^2 = 6 \times 64 = 384 \, \text{cm}^2$

So, the total surface area of the cube is $384 \, \text{cm}^2$.

Would you like further details on the steps or have any questions?

Here are 5 related questions to expand on this topic:

1. What is the length of the diagonal of a face of the cube?
2. How does the surface area change if the side length is doubled?
3. What is the volume of a cube with a surface area of 150 cm²?
4. If the volume of the cube is tripled, how does the side length change?
5. How do you find the surface area of a rectangular prism?

Tip: The surface area of a cube grows much faster than its volume as you increase the side length, since surface area depends on $s^2$ while volume depends on $s^3$.