What are the tightest known bounds for the number of groups of order $2048$? 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 ?

 A: 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.
A: I believe the subset of groups you're thinking of is the set of $p$-class 2 groups of order $p^n$. Higman uses this to obtain his lower bound which is 
$$ p^{\frac{2}{27}(n^3-6n^2)}.$$
Thus, a lower bound on the number of groups of order $2^{11}$ is $2^{\frac{1210}{27}}\approx 3\times 10^{13}$.
In that same paper, Higman computes an easy upper bound of 
$$p^{\frac{1}{6}(n^3-n)},$$
which gives us an upper bound of $2^{220}\approx 10^{66}$.
The upper bound here is not the best known (even Higman gives a slightly better asymptotic bound in the same paper), but I think the lower bound might be the best we have.
