The construction is called "isotopy" and you ask for quasigroups that are not "isotopic to groups".

It's been proved a long time ago that if a loop (= a quasigroup with a unit element) is isotopic to a group G, it is actually isomorphic to G. Consequently, any non-associative loop is an answer to your question. The smallest example has four elements.

The same theorem gives a simple criterion to recognize quasigroups isotopic to a group. Given a quasigroup with an operation , pick an element e and form a new quasigroup, by x # y = x/e * e\y. This is a loop, where ee is the unit element. If # is associative, then, obviously, * is isotopic to a group. If # is not associative, the theorem assures no group isotope can be found.

The construction is called "isotopy" and you ask for quasigroups that are not "isotopic to groups".

It's been proved a long time ago that if a loop (= a quasigroup with a unit element) is isotopic to a group G, it is actually isomorphic to G. Consequently, any non-associative loop is an answer to your question. The smallest example has five elements, and is unique (up to isotopy):

0 1 2 3 4          
1 0 3 4 2            
2 3 4 1 0            
3 4 0 2 1    
4 2 1 0 3

The same theorem gives a simple criterion to recognize quasigroups isotopic to a group. Given a quasigroup with an operation , pick an element e and form a new quasigroup, by x # y = x/e * e\y. This is a loop, where ee is the unit element. If # is associative, then, obviously, * is isotopic to a group. If # is not associative, the theorem assures no group isotope can be found.