A heavy-fermion superconductor is a unique spin-triplet superconductor with multiple superconducting phases. Here, we provide the first report on a first-principles analysis of the microscopic superconducting gap structure. We find that the promising gap structure is an unprecedented state, which is completely different from the previous phenomenological models. Our obtained state has in-plane twofold vertical line nodes on small Fermi surfaces and point nodes with linear dispersion on a large Fermi surface. These peculiar features cannot be explained in the conventional spin representation, but is described by the group-theoretical representation of the Cooper pairs in the total angular momentum space. Our findings shed new light on the long-standing problems in the superconductivity of .