Olympiad Geometry · Superchallenge problem

Cube $ABCDEFGH$ has edge $6$, and $\overline{AG}$ is one of its space diagonals. A plane perpendicular to $\overline{AG}$ crosses it at the point $P$ with $AP : PG = 5 : 4$. Find the area of the cross-section of the cube cut by this plane. Give an exact answer.