Affine toric varieties with an open orbit of an $SL_n$ action

We study affine toric varieties with an action of group $SL_n$ with a dense orbit. A characterisation in terms of $SL_n \times Q$-modules is given where $Q$ is a quasitorus. This characterisation is more explicitly expanded in case $n=3$. It is shown that in case $n=3$ the class group rank is not greater than $3$, however it is unbounded when $n\geq 4$.