  Groups with commuting inner mappings are of nilpotency class at most two, but
there exist loops with commuting inner mappings and of nilpotency class higher
than two, called loops of Cs\"org\H{o} type. In order to obtain small loops of
Cs\"org\H{o} type, we expand our programme from `Explicit constructions of
loops with commuting inner mappings', European J. Combin. 29 (2008), 1662-1681,
and analyze the following setup in groups:
  Let $G$ be a group, $Z\le Z(G)$, and suppose that $\delta:G/Z\times G/Z\to Z$
satisfies $\delta(x,x)=1$, $\delta(x,y)=\delta(y,x)^{-1}$,
$z^{yx}\delta([z,y],x) = z^{xy}\delta([z,x],y)$ for every $x$, $y$, $z\in G$,
and $\delta(xy,z) = \delta(x,z)\delta(y,z)$ whenever $\{x,y,z\}\cap G'$ is not
empty.
  Then there is $\mu:G/Z\times G/Z\to Z$ with $\delta(x,y) =
\mu(x,y)\mu(y,x)^{-1}$ such that the multiplication $x*y=xy\mu(x,y)$ defines a
loop with commuting inner mappings, and this loop is of Cs\"org\H{o} type (of
nilpotency class three) if and only if $g(x,y,z) =
\delta([x,y],z)\delta([y,z],x)\delta([z,x],y)$ is nontrivial.
  Moreover, $G$ has nilpotency class at most three, and if $g$ is nontrivial
then $|G|\ge 128$, $|G|$ is even, and $g$ induces a trilinear alternating form.
We describe all nontrivial setups $(G,Z,\delta)$ with $|G|=128$. This allows us
to construct for the first time a loop of Cs\"org\H{o} type with an inner
mapping group that is not elementary abelian.
