"If it is a "8x51", what cartridge is that? Based on the .308 Winchester or what? "

very funny, the 8x51 was probably designed before the designer of the 308 win were born.
its a shortened 8x57 for smaller game like roe deer to make the rifle smaller and lighter.
good thing today you can make brass very easy from 308 win.
you only resize the 308 brass and its ready.