Solution:

To find the dimensions of the gravitational constant $G$, use Newton's law of gravitation:

$$F = \frac{G M_1 M_2}{r^2}$$

Where:
- $F$ is force (with dimensions $[M L T^{-2}]$),
- $M_1$ and $M_2$ are masses (with dimensions $[M]$),
- $r$ is distance (with dimensions $[L]$).

Rearranging for $G$:

$$G = \frac{F r^2}{M_1 M_2}$$

Substitute the dimensions:

$$G = \frac{[M L T^{-2}] [L^2]}{[M][M]}$$

Simplify:

$$G = [M^{-1} L^3 T^{-2}]$$

Thus, the dimensions of $G$ are $[M^{-1} L^3 T^{-2}]$.