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We claim that the class of $G$ is not bounded. Let $G$ be a $3$-group of maximal class and order $3^n>3^4$ containing no abelian subgroup of index $3$. Then $G$ has a metacyclic subgroup $M$ of index $3$ such that $|M'|=3$. It follows that $M$ is minimal nonabelian so that $|M:\text{Z}(M)| =3^2$. If $x\in M-\text{Z}(M)$, then $\text{C}_G(x)=\langle x,\text{Z}(M)\rangle$ is a subgroup of $M$ of index $3$. It is known that $|G|$ is not bounded so the same holds for $\text{cl}(G)$. But it is not known if the derived lenghth of $G$ is bounded.

We claim that the class of $G$ is not bounded. Let $G$ be a $3$-group of maximal class and order $3^n>3^4$ containing no abelian subgroup of index $3$. Then $G$ has a metacyclic subgroup $M$ of index $3$ such that $|M'|=3$. It follows that $M$ is minimal nonabelian so that $|M:\text{Z}(M)| =3^2$. If $x\in M-\text{Z}(M)$, then $\text{C}_G(x)=\langle x,\text{Z}(M)\rangle$ is a subgroup of $M$ of index $3$. It is known that $|G|$ is not bounded so the same holds for $\text{cl}(G)$. But it is not known if the derived length of $G$ is bounded.

We claim that the class of $G$ is not bounded. Let $G$ be a $3$-group of maximal class and order $3^n>3^4$ containing no abelian subgroup of index $3$. Then $G$ has a metacyclic subgroup $M$ of index $3$ such that $|M'|=3$. It follows that $M$ is minimal nonabelian so that $|M:\text{Z}(M)| =3^2$. If $x\in M-\text{Z}(M)$, then $\text{C}_G(x)=\langle x,\text{Z}(M)\rangle$ is a subgroup of $M$ of index $3$. It is known that $|G|$ is not bounded so the same holds for $\text{cl}(G)$.

We claim that the class of $G$ is not bounded. Let $G$ be a $3$-group of maximal class and order $3^n>3^4$ containing no abelian subgroup of index $3$. Then $G$ has a metacyclic subgroup $M$ of index $3$ such that $|M'|=3$. It follows that $M$ is minimal nonabelian so that $|M:\text{Z}(M)| =3^2$. If $x\in M-\text{Z}(M)$, then $\text{C}_G(x)=\langle x,\text{Z}(M)\rangle$ is a subgroup of $M$ of index $3$. It is known that $|G|$ is not bounded so the same holds for $\text{cl}(G)$. But it is not known if the derived lenghth of $G$ is bounded.

We claim that the class of $G$ is not bounded. Let $G$ be a $3$-group of maximal class and order $3^n>3^4$ containing no abelian subgroup of index $3$. Then $G$ has a metacyclic subgroup $M$ of index $3$ such that $|M'|=3$. It follows that $M$ is minimal nonabelian so that $|M:\text{Z}(M)| =3$. If $x\in M-\text{Z}(M)$, then $\text{C}_G(x)=\langle x,\text{Z}(M)\rangle$ is a subgroup of $M$ of index $3$. It is known that $|G|$ is not bounded so the same holds for $\text{cl}(G)$.

We claim that the class of $G$ is not bounded. Let $G$ be a $3$-group of maximal class and order $3^n>3^4$ containing no abelian subgroup of index $3$. Then $G$ has a metacyclic subgroup $M$ of index $3$ such that $|M'|=3$. It follows that $M$ is minimal nonabelian so that $|M:\text{Z}(M)| =3^2$. If $x\in M-\text{Z}(M)$, then $\text{C}_G(x)=\langle x,\text{Z}(M)\rangle$ is a subgroup of $M$ of index $3$. It is known that $|G|$ is not bounded so the same holds for $\text{cl}(G)$.