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Let be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of inducing the minimal gradation on . The corresponding minimal -algebra introduced by Kac and Wakimoto has strong generators in weights , and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra , where denotes the centralizer of . Therefore, has an action of a connected Lie group with Lie algebra , where denotes the even part of . We show that for any reductive subgroup , and for any reductive Lie algebra , the orbifold and the coset are strongly finitely generated for generic values of . Here denotes the affine vertex algebra associated to . We find explicit minimal strong generating sets for when and is either , , for ,  …