Mach Numbers
For a standard atmosphere (1959 ARDC, 1962 ICAO, 1976 NOAA/NASA/USAF), temperature at sea level is 288.16 K, the tropopause is at 11 km altitude, and temperature decays linearly at 6.5 K/km to 216.66 K at the tropopause.
The speed of sound in knots is about 39 times the square root of the absolute temperature in degrees Kelvin.
Hence the speed of sound for a standard atmosphere is about 661 kts at sea level and about 573 kts at the tropopause (11 km, or about 36,000 ft).
Speeds of 323 kts to 350 kts are about 0.49 M to 0.53 M at sea level, and about 0.56 M to 0.61 M at high altitude.
These are all well below max range speed for a 777-200ER, and pretty far below max endurance speed.