Open-shell singlet nature, aromaticity, and second hyperpolarizabilities of two kinds of curved π-conjugated fragments of C60 fullerene, called C60 flakes, together with their corresponding planar analogues are theoretically investigated. It is found that one curved system, A-1, exhibits intermediate open-shell singlet nature, while the other structure, B-1, and the two corresponding planar systems possess a closed-shell ground state. The variation of the open-shell singlet nature is explained by the number of Clar's sextets in the resonance structures, which is associated with the local aromaticity, as quantified by the NICS0 values. The calculated orientationally averaged second hyperpolarizabilities show that the intermediate open-shell singlet system A-1 exhibits the best performance, and that the other curved system B-1 is the worst among the four C60 flakes, whereas all systems exhibit better performance than the original spherical C60 system. This tendency suggests that a lower dimensionality in the π-conjugated structure is beneficial to the construction of efficient nonlinear optical molecules. The present results not only show that the open-shell singlet system A-1 is a performant nonlinear optical molecule, but they also contribute to a deeper understanding of the electronic structure of curved π-conjugated molecules.