We present the precision measurement from May 2011 to May 2017 (79 Bartels rotations) of the proton fluxes at rigidities from 1 to 60 GV and the helium fluxes from 1.9 to 60 GV based on a total of 1 x 109 events collected with the Alpha Magnetic Spectrometer aboard the International Space Station. This measurement is in solar cycle 24, which has the solar maximum in April 2014. We observed that, below 40 GV, the proton flux and the helium flux show nearly identical fine structures in both time and relative amplitude. The amplitudes of the flux structures decrease with increasing rigidity and vanish above 40 GV. The amplitudes of the structures are reduced during the time period, which started one year after solar maximum, when the proton and helium fluxes steadily increase. Above similar to 3 GV the p/He flux ratio is time independent. We observed that below similar to 3 GV the ratio has a long-term decrease coinciding with the period during which the fluxes start to rise.