We have used scanning electron microscopy with polarization analysis and photoemission electron microscopy to image the two-dimensional magnetization of permalloy films patterned into Penrose P2 tilings (P2T). The interplay of exchange interactions in asymmetrically coordinated vertices and short-range dipole interactions among connected film segments stabilize magnetically ordered, spatially distinct sublattices that coexist with frustrated sublattices at room temperature. Numerical simulations that include long-range dipole interactions between sublattices agree with images of as-grown P2T samples and predict a magnetically ordered ground state for a two-dimensional quasicrystal lattice of classical Ising spins.