I will call this construction a chapel, for lack of a better name. This is no more than a relatively big dome with its corresponding support structure. The dome is made of eight independent sections, each one of them made of five pentagonal prisms and four cubes.
Although the official minimum angle between two Geomag rods attached to the same ball is 60 degrees, in this case each two consecutive prisms join at an angle of 54 degrees (opposite to the joining cube: 360 – 90 – 2 × 108). The rods forming the angle touch one another, but there seems not to be loss of adherence, or any other problem.
All eight sections are joined by square pyramids to a top central piece with an octagonal profile (here it is a square gyrobicupola but it also could be a orthobicupola):
Each section rests on a column made of square antiprisms, and with a double-scale square as a foot. These feet are placed in the external wedge-shaped gaps left between eight regular octagons when placed in a circle, as shown in the following picture, which includes sketches for the plan (gray hues), and for the dome profile (bluish hue):
The following picture show a possible auxiliary structure (red rods and blue panels) to help placing the columns as in the previous sketch: