  We study the supersymmetric quantum dynamics of the cosmological models
obtained by reducing $D=5$ supergravity to one timelike dimension. This
consistent truncation has fourteen bosonic degrees of freedom, while the
quantization of the homogeneous gravitino field leads to a
$2^{16}$--dimensional fermionic Hilbert space. We construct a consistent
quantization of the model in which the wave function of the Universe is a
$2^{16}$--component spinor %\textcolor{red}{of Spin(24,8)} depending on
fourteen continuous coordinates, which satisfies eight Dirac-like wave
equations (supersymmetry constraints) and one Klein-Gordon-like equation
(Hamiltonian constraint). The fermionic part of the quantum Hamiltonian is
built from operators that generate a $2^{16}$-dimensional representation of the
(infinite-dimensional) maximally compact sub-algebra $K(G_2^{++})$ of the
rank-4 hyperbolic Kac--Moody algebra $G_2^{++}$. The (quartic-in-fermions)
squared-mass term $\widehat \mu^2$ entering the Klein-Gordon-like equation has
several remarkable properties: (i) it commutes with the generators of
$K(G_2^{++})$; and (ii) it is a quadratic polynomial in the fermion number $N_F
\sim \overline\Psi \Psi$, and a symplectic fermion bilinear $C_F \sim \Psi
C\Psi$. Some aspects of the structure of the solutions of our model are
discussed, and notably the Kac-Moody meaning of the operators describing the
reflection of the wave function on the fermion-dependent potential walls
("quantum fermionic Kac-Moody billiard").
