  We study the geometry of the space of Mermin pentagrams, objects that are
used to rule out the existence of noncontextual hidden variable theories as
alternatives to quantum theory. It is shown that this space of 12096 possible
pentagrams is organized into 1008 families, with each family containing a
"double six" of pentagrams. The 1008 families are connected to a special set of
Veldkamp lines in the Veldkamp space of three qubits an object well-known to
finite geometers but has only been introduced to physics recently. Due to the
transitive action of the symplectic group on this set of Veldkamp lines it is
enough to study only one "canonical" double six configuration of pentagrams. We
prove that the geometry of this double-six configuration is encapsulated in the
weight diagram of the 20 dimensional irreducible representation of the group
$SU(6)$. As an interesting by-product of our approach we show that Mermin
pentagrams and a class of Mermin squares labelled by three qubit Pauli
operators are inherently related. We conjecture that by studying the
representation theoretic content of other Veldkamp lines of the Veldkamp space
for $N$-qubits makes it possible to find new contextual configurations in a
systematic manner.
