There are four different shapes of quadrilaterals here, but they have one thing in common: there are twelve of each of them. Add those 48 quads to the twelve regular heptagons, and that gives a total of 60 faces. I found this polyhedron while playing with the snub dodecahedron, using Stella 4d: Polyhedron Navigator, a program you can try for free right here.

In the image below, each of the four types of quadrilateral appears with its own color.

I made this using Stella 4d, which you can try for free right here.

I made this using Stella 4d, which you can try for free at this website.

In addition to the four regular heptagons, this polyhedron has four quadrilateral faces, plus 24 triangles. I made it using Stella 4d, which you can try for free here. The second image, below, shows the same polyhedron from a different angle.

This polyhedron has 230 faces, and I made it using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

In addition to the sixty heptagons, there are twelve regular pentagons and sixty quadrilaterals in this polyhedron, along with 380 triangles of various shapes and sizes. That’s 512 faces in all.

I used Stella 4d: Polyhedron Navigator to make this. You may try this program, for free, at http://www.software3d.com/Stella.php.