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Citation: Brown, Benjamin J., et al. "Poking holes and cutting corners to achieve Clifford gates with the surface code." Physical Review X 7.2 (2017): 021029.
There are lots of different models for quantum computation with a given topological order. The general scheme for the toric code is that Pauli gates are very easy, Clifford gates are possible but a bit harder, and everything else requires non-topological action. This paper talks a bit about the various approaches to implement the Clifford group.
This suggests that, in a higher-order generalization, we should expect there to be multiple ways of arriving at the same gate set. Namely, there should be multiple ways of getting to the same answer of "fault tolerant gates are the generalized Clifford group". This is a good article for the well-known surface code case.