  We study the critical behavior of a generalized icosahedral model on the
simple cubic lattice. The field variable of the icosahedral model might take
one of twelve vectors of unit length, which are given by the normalized
vertices of the icosahedron, as value. Similar to the Blume-Capel model, where
in addition to $-1$ and $1$, as in the Ising model, the spin might take the
value $0$, we add in the generalized model $(0,0,0)$ as allowed value. There is
a parameter $D$ that controls the density of these voids. For a certain range
of $D$, the model undergoes a second-order phase transition. On the critical
line, $O(3)$ symmetry emerges. Furthermore, we demonstrate that within this
range, similar to the Blume-Capel model on the simple cubic lattice, there is a
value of $D$, where leading corrections to scaling vanish. We perform Monte
Carlo simulations for lattices of a linear size up to $L=400$ by using a hybrid
of local Metropolis and cluster updates. The motivation to study this
particular model is mainly of technical nature. Less memory and CPU time are
needed than for a model with $O(3)$ symmetry at the microscopic level. As the
result of a finite-size scaling analysis we obtain $\nu=0.71164(10)$,
$\eta=0.03784(5)$, and $\omega=0.759(2)$ for the critical exponents of the
three-dimensional Heisenberg universality class. The estimate of the irrelevant
renormalization group eigenvalue that is related with the breaking the $O(3)$
symmetry is $y_{ico}=-2.19(2)$.
