We demonstrated the synchronized high-power ytterbium-doped fiber amplification of the near-infrared spectral fraction of a few-cycle Ti:sapphire laser at 79.5-MHz repetition rate, which was experimentally realized with a two-stage ytterbium-doped fiber pre-amplifier and a two-stage double-clad fiber power amplifier in cascade. An overall power amplification up to 86.1 dB was achieved at 1030 nm with sufficient suppression of amplified spontaneous emission noises, achieving to the best of our knowledge the highest synchronized laser pulses of 262 W in average power. The amplified high-power pulses were demonstrated to maintain good temporal synchronization with the few-cycle signal pulses.