The well-developed class of the quantum error correction codes is known as “additive codes”, which arise as subspaces stabilized by Abelian subgroups of the Pauli group. In this paper several codes of the much less investigated “non-additive” class are discussed. Namely, these are the binary codes associated with the action of non-Abelian group. More precisely, the authors concentrate their attention on the codes on which the symmetrie group acts trivially and call it permutationally invariant codes. These codes require at least 7 qubits to correct all one-bit errors. Authors found two distinct 7-bit codes of this type. Also they investigated a large family of permutationally invariant 9-bit codes and found boundaries of its power in the area of two-bit error correction. It was shown that no 9-bit code can correct all two-bit errors of the form \(Z_jZ_k\) and all one-bit errors of the form \(X_k\) and \(Z_k\) at the same time, where \(X_i\) and \(Z_i\) denote the action of the \(\sigma_x\) and \(\sigma_z\) Pauli spin operators. Also it was proved that if one apply a slightly modified restriction on the simultaneously one-bit and two-bit error correction, which require to correct all errors of the form \(X_k\) and all errors of the form \(X_jX_k\) and \(Z_jZ_k\), it can be achieved using the simple 5-bit repetition code.