The number of groups of order $2048$ is unknown. 

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

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