Turing degrees of nonabelian groups

For a countable structure $\mathcal{A}$, the (Turing) degree spectrum of $\mathcal{A}$ is the set of all Turing degrees of its isomorphic copies. If the degree spectrum of $\mathcal{A}$ has the least degree $\mathbf{d}$, then we say that $\mathbf{d}$ is the (Turing) degree of the isomorphism type of $\mathcal{A}$. So far, degrees of the isomorphism types have been studied for abelian and metabelian groups. Here, we focus on highly nonabelian groups. We show that there are various centerless groups whose isomorphism types have arbitrary Turing degrees. We also show that there are various centerless groups whose isomorphism types do not have Turing degrees.