Are you running computer experiments to verify conjectures? If so, the GAP SmallGroups library will give you exactly what you want up to $n = 1023$.

For example, the GAP commands

  n:=16;; G:=SmallGroup(n, Random(1,NumberSmallGroups(n)));

will return you a group chosen uniformly at random from the groups of order $n=16$. Similar commands will work up to $n=1023$. Indeed, it will work for also for all orders up to 2000 except for 1024, and for a considerable number of other orders.

You might also find the SmallGroups library web page to be helpful:
https://gap-packages.github.io/smallgrp/doc/chapBib_mj.html

It describes some of the methods involved. If you're willing to select a group uniformly at random from a subcollection of all the groups of order $2^k$, then there are several papers on groups of order $2^k$ (for varying values of $k$) cited there. Applying the methods therein might be enough for you, depending on what your specific needs are.

Are you running computer experiments to verify conjectures? If so, the GAP SmallGroups library will give you exactly what you want up to $n = 1023$.

For example, the GAP commands

  n:=16;; G:=SmallGroup(n, Random(1,NumberSmallGroups(n)));

will return you a group chosen uniformly at random from the groups of order $n=16$. Similar commands will work up to $n=1023$. Indeed, it will work for also for all orders up to 2000 except for 1024, and for a considerable number of other orders.

You might also find the SmallGroups library web page to be helpful.

It describes some of the methods involved. If you're willing to select a group uniformly at random from a subcollection of all the groups of order $2^k$, then there are several papers on groups of order $2^k$ (for varying values of $k$) cited there. Applying the methods therein might be enough for you, depending on what your specific needs are.

Are you running computer experiments to verify conjectures? If so, the GAP SmallGroups library will give you exactly what you want up to $n = 1023$.

For example, the GAP commands

  n:=16;; G:=SmallGroup(n, Random(1,NumberSmallGroups(n)));

will return you a group chosen uniformly at random from the groups of order $n=16$. Similar commands will work up to $n=1023$. Indeed, it will work for also for all orders up to 2000 except for 1024, and for a considerable number of other orders.

You might also find the SmallGroups library web page to be helpful:
http://www.icm.tu-bs.de/ag_algebra/software/small/

It describes some of the methods involved. If you're willing to select a group uniformly at random from a subcollection of all the groups of order $2^k$, then there are several papers on groups of order $2^k$ (for varying values of $k$) cited there. Applying the methods therein might be enough for you, depending on what your specific needs are.

Are you running computer experiments to verify conjectures? If so, the GAP SmallGroups library will give you exactly what you want up to $n = 1023$.

For example, the GAP commands

  n:=16;; G:=SmallGroup(n, Random(1,NumberSmallGroups(n)));

will return you a group chosen uniformly at random from the groups of order $n=16$. Similar commands will work up to $n=1023$. Indeed, it will work for also for all orders up to 2000 except for 1024, and for a considerable number of other orders.

You might also find the SmallGroups library web page to be helpful:
https://gap-packages.github.io/smallgrp/doc/chapBib_mj.html

It describes some of the methods involved. If you're willing to select a group uniformly at random from a subcollection of all the groups of order $2^k$, then there are several papers on groups of order $2^k$ (for varying values of $k$) cited there. Applying the methods therein might be enough for you, depending on what your specific needs are.