Moment of inertia of a square plate of side $l$ about the axis passing through one of the corner and perpendicular to the plane of square plate is given by:

Use parallel axis theorem to get the magnitude of the moment of inertia of a square plate of side $l$ about the axis passing through one of the corners and perpendicular to the plane of a square plate, 

$I_z=I_{\mathrm{cm}}+M{\left(\frac{l}{\sqrt{2}}\right)}^2$

$\Rightarrow I_z=\frac{M{l}^2}{6}+\frac{M{l}^2}{2}=\frac{4M{l}^2}{6}=\frac{2M{l}^2}{3}$.