This fullerene contains a single C2 axis which passes through the center of a six-membered ring on one side and the midpoint of a bond on the other side. According to Fu-3.1.2 the ring is preferred for beginning the numbering. Because of symmetry there are only three pathways to be considered, namely clockwise a to b to a', b to a' to a'', and a' to a'' to b' (Fig. 5a). These lead to the contiguous helical numberings shown in Fig. 5b, 5c, and 5d, respectively. By application of Fu-3.1.3 the numbering of Fig. 5c is preferred because it terminates at an atom belonging to the bond bisected by the C2 axis, whereas the other numberings terminate at an atom one (Fig. 5d) and two (Fig. 5b) bonds removed from the C2 axis, respectively.
Fig. 5. Systematic numbering of (C78-C2v)[5,6]fullerene
5. P.W. Fowler and D.E. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, 1995