I have managed to make the chiral isomer of T240 (D5h) by shifting the pentagons on the outer rim by one bead. Unfortunately, I was unable to put the beads into a complete torus due to the huge strain energy for this structure. The stable chiral forms of T240 originating from the isomer with D5h point group do not exist! It is quite amazing it is possible to get a first order approximation for the geometry and stability of carbon spheroids or toroids by using molecular beadings. But the price to pay is that you need to spend several hours on the beadings for a single structure.
Previously, I have made two D5h T240 which can be converted into each other by Stone-Wales transformation. But, according to Ihara's paper, there seems to be another kind of T240 isomer which can be generated by moving ten pentagons on the outer rim to the mid-edge position (i.e. shifting 36 degree about z-axis). I am not sure about the stability of this structure. But since the outer-rim pentagons and inter-rim heptagons are arranged in the staggered pattern, I suspect the strain energy for this type of structure is large.