Shorthand Adjectives:

Generation X was the first alphanumerically-identified generation, so they should lay claim to the nicknames "GenXers" or "Xers."

As far as the generations that succeed Gen X:

Gen Y = "Millennials" or "iGens"

Gen Z = "Centennials" or "Zoomers"

Gen AA = "Alphas" or "NuAlphas" (most of them haven't been born yet, so we'll need them to get older before more solid nicknames arise)