Can a commutative block cipher be indistinguishable from a random permutation, for fixed key? Define a commutative block cipher with keyspace the finite set $K$, and data space the finite set $S$, to be an application
$$\begin{align}
E:K\times S&amp;\mapsto S\\
(k,x)&amp;\mapsto E(k,x)\text{ also noted }E_k(x)\\
\text{such that }&amp;\forall k\in K,\forall x\in S, \forall y\in S,\text{ if }E(k,x)=E(k,y)\text{ then }x=y\\
\text{and }&amp;\forall k\in K,\forall k'\in K, \forall x\in S,E(k',E(k,x))=E(k,E(k',x))\\
\end{align}$$
Note: the first property states that the application $E_k$ from $S$ to $S$ is injective, implying that it is a permutation of $S$ given this is a finite set, and that's standard for a cipher; the second property is what makes the cipher commutative. 

Question: what commutative block cipher do we have that is


conjectured computationally indistinguishable with effort polynomial in $\log(|K|)$ from a random permutation for unknown random fixed key $k$, assuming the attacker can obtain ciphertext for iteratively chosen plaintext;
efficiently computable, at least in the encryption direction (and preferably for decryption as well)?


If there was none: do we have some proof this can't be achieved?



In this answer, I explored a variant of the Pohlig-Hellman Exponentiation Cipher $(k,x)\mapsto E_k(x)=x^k\pmod p$ but stalled with something that remains distinguishable from a random permutation due to a multiplicative property similar to that of naked RSA: $\forall k\in K,\forall x\in S, \forall y\in S, E_k(x\cdot y\bmod p)=E_k(x)\cdot E_k(y)\bmod p$.



Update: the question as is was perfectly answered by poncho. Here is take two.
It can't be achieved under the assumptions you are making, because the attacker can distinguish it by selecting an arbitrary $k'$, and checking if $E(k')$ commutes with the permutation in question.  That is, to check a permutation $P$, we pick an arbitrary $x$, and check if:

$E(k', P(x)) = P(E(k',x))$

This equation always holds if $P = E(k)$ for some value $k$, and rarely holds if $P$ is a random permutation.