Counterexample to a conjecture about braces

We find an example of a finite solvable group (in fact, a finite p-group) in which is not possible to define a left brace structure or, equivalently, which is not an IYB group. This answers a question posed by Cedó, Jespers and del Río related to the Yang-Baxter equation. Our argument is an improvement of an argument of Rump, using results about Hopf-Galois extensions and LSA structures. We explain explicitly the relation between these two areas of mathematics and brace theory, hoping that it will be useful in the future.