We compute the class of the classifying stack of the exceptional algebraic group
G2
and of the spin groups
Spin7
and
Spin8
in the Grothendieck ring of stacks, and show that they are equal to the inverse of the class of the corresponding group. Furthermore, we show that the computation of the motivic classes of the stacks
ℬSpinn
can be reduced to the computation of the classes of
ℬΔn
, where
Δn⊂Pinn
is the “extraspecial
2
-group”, the preimage of the diagonal matrices under the projection
Pinn→On
to the orthogonal group.