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The gnu (or Group NUmber) function describes how many groups there are of a given order. The number of groups of each order are known up to 2047, see https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf Has any progress been made on the number of groups of order 2048? This case is particularly difficult due to 2048 being a large power of 2. It is known that the number of groups of order 2048 of nilpotency class 2 is 1,774,274,116,992,170 (according to the above link), and apparently the full group number is expected to agree with this number in the first three digits.

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No, it is unknown, and I don't think we will find it anytime soon. For the state of the art, see our 2017 paper "Constructing groups of ‘small’ order: Recent results and open problems" DOI (here is a PDF). I collected the known data on a little website for easier browsing. And am working as I type this on packaging it up for GAP.

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