  The zero-temperature properties of the SO(5) bilinear-biquadratic Heisenberg
spin chain are investigated by means of a low-energy approach and large scale
numerical calculations. In sharp contrast to the spin-1 SO(3) Heisenberg chain,
we show that the SO(5) Heisenberg spin chain is dimerized with a two-fold
degenerate ground state. On top of this gapful phase, we find the emergence of
a non-degenerate gapped phase with hidden (Z$_2$ $\times$ Z$_2$)$^2$ symmetry
and spin-3/2 edge states that can be understood from a SO(5) AKLT wave
function. We derive a low-energy theory describing the quantum critical point
which separates these two gapped phases. It is shown and confirmed numerically
that this quantum critical point belongs to the SO(5)$_1$ universality class.
