We consider a new type of frustrated spin network with pentagonal loops and long-range quasiperiodic structural order. Five-fold loops can be expected to occur naturally in quasicrystals, and experimental studies of icosahedral alloys show manifestations of local five-fold symmetry in a variety of different physical contexts. Our model considers classical spins placed on the vertices of a subtiling of the two-dimensional Penrose tiling, and interacting with nearest neighbours via antiferromagnetic bonds. The ground state of this fractal system has a complex magnetic structure, and is studied analytically as well by Monte Carlo simulation on finite clusters.