AbstractWe study $$\kappa $$ κ -maximal cofinitary groups for $$\kappa $$ κ regular uncountable, $$\kappa = \kappa ^{<\kappa }$$ κ = κ < κ . Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell’s theorem, we show that: Any $$\kappa $$ κ -maximal cofinitary group has $${<}\kappa $$ < κ many orbits under the natural group action of $$S(\kappa )$$ S ( κ ) on $$\kappa $$ κ . If $$\mathfrak {p}(\kappa ) = 2^\kappa $$ p ( κ ) = 2 κ then any partition of $$\kappa $$ κ into less than $$\kappa $$ κ many sets can be realized as the orbits of a $$\kappa $$ κ -maximal cofinitary group. For any regular $$\lambda > \kappa $$ λ > κ it is consistent that there is a $$\kappa $$ κ -maximal cofinitary group which is universal for groups of size $${<}2^\kappa = \lambda $$ < 2 κ = λ . If we only require the group to be universal for groups of size $$\kappa $$ κ then this follows from $$\mathfrak {p}(\kappa ) = 2^\kappa $$ p ( κ ) = 2 κ .