  The face poset of the permutohedron realizes the combinatorics of linearly
ordered partitions of the set $[n]=\{1,...,n\}$. Similarly, the
cyclopermutohedron is a virtual polytope that realizes the combinatorics of
cyclically ordered partitions of the set $[n+1]$. The cyclopermutohedron was
introduced by the third author by motivations coming from configuration spaces
of polygonal linkages. In the paper we prove two facts: (1) the volume of the
cyclopermutohedron equals zero, and (2) the homology groups $H_k$ for
$k=0,...,n-2$ of the face poset of the cyclopermutohedron are non-zero free
abelian groups. We also present a short formula for their ranks.
