  We present an exact diagonalization study of the quantum Heisenberg
antiferromagnet on the fractal Sierpinski gasket for spin quantum numbers
s=1/2,s=1 and s=3/2. Since the fractal dimension of the Sierpinski gasket is
between one and two we compare the results with corresponding data of one- and
two-dimensional systems. By analyzing the ground-state energy, the low-lying
spectrum, the spin-spin correlation and the low-temperature thermodynamics we
find arguments, that the Heisenberg antiferromagnet on the Sierpinski gasket is
probably disordered not only in the extreme quantum case s=1/2 but also for s=1
and s=3/2. Moreover, in contrast to the one-dimensional chain we do not find a
distinct behavior between the half-integer and integer-spin Heisenberg models
on the Sierpinski gasket. We conclude that magnetic disorder may appear due to
the interplay of frustration and strong quantum fluctuations in this spin
system with spatial dimension between one and two.
