TITLE: What are the tightest known bounds for the number of groups of order $2048$?
QUESTION [8 upvotes]: The number of groups of order $2048$ is unknown. 

What are the tightest known bounds (lower and upper bounds : I am interested in both) for the number of groups of order $2048$ ?

I know the asymptotic formula for $p^k$, but I do not think that it gives a useful bound for $p^k=2048$. Somewhere, I read that a subset of the groups (but I do not remember what kind of subset) was calculated to obtain a lower bound. 

Can it be estimated how long it will take to determine the number ?

REPLY [3 votes]: According to the second paragraph of 
https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf
(which I posted in another of your questions yesterday)
"[The number of groups of order 2048] is still not precisely known, but it strictly exceeds 1774274116992170, which is the exact number
of groups of order 2048 that have exponent-2 class 2, and can confidently be expected to agree with that number in its first 3 digits.
"
This is probably the best answer you can get easily.