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$gnu(n)$ denotes the number of groups of order $n$ upto isomorphy.

What is the smallest number $n$ with $gnu(n)=2016$ ?

The numbers upto $n=2047$ do not satisfy the given property. I do not know any value upto $50,000$ satisfying the property, but of course, there might be one.

The smallest square-free number with $gnu(n)=2016$ is $n=1,607,970=2\cdot3\cdot5\cdot7\cdot13\cdot19\cdot31$

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