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.



Has gnu(2048) been found?
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](http://dx.doi.org/10.1007/978-3-319-70566-8_8) (here is a [PDF](https://www.quendi.de/data/papers/EHH2018-small-groups.pdf)). I collected the known data on [a little website](http://groups.quendi.de/) for easier browsing. And am working as I type this on packaging it up for GAP.


