Over the last two pages, I notice that Mach Speeds start at:
FLs 240, 250, 260 and the last post, 290.
From FAAH 7110.655-7-1 Speed Adjustmente.
Express speed adjustments in terms of knots based on indicated airspeed (IAS) in 10-knot increments. At or above FL 240, speeds may be expressed in terms of Mach numbers in 0.01 increments for turbojet aircraft with Mach meters
(i.e., Mach 0.69, 0.70, 0.71, etc.).
As for what the Center guys actually use...you'll have to ask them.