This polyhedron can be made in 5 colors. Use three of the colors for your octagons (make 2 of each color), 1 color for your hexagons (make 8) and another color for your squares (make 12).
To see the full-sized template click on this image:
Begin by surrounding one octagon alternately with squares and hexagons. These 9 pieces will form your beginning bowl of the polyhedron.
As usual, opposites sides of this polyhedron will contain parallel faces of the same colors. So, your octagons will always be opposite an octagon of the same color.
On opposite sides of the bowl attach two more pair of octagons of the same colors. These side octagons are attached to the squares. Now you can see that these side octagons will be surrounded alternately by squares and hexagons and you can finish the assembly.