What are the groups for NxNxN puzzle cubes called?

[The Bill]

The group of moves for the 3x3x3 puzzle cube is the Rubik’s Cube group: https://en.wikipedia.org/wiki/Rubik's_Cube_group.

What are the groups of moves for NxNxN puzzle cubes called in general? Is there even a standardized term?

I've been trying to find literature on the groups for the 2x2x2, 4x4x4, and 5x5xx5 puzzle cubes, but all I keep getting is more about the 3x3x3's Rubik’s Cube group.

 

[fresh_42]

At least the Wikipedia page describes a way how to determine them.

I've found a Bachelor paper about them. The author calls them Magic Cubes $M_n$. He mainly proves complexity statements such as God's number for $M_3$ is less than $30$ and for $M_2$ it equals $14$ and considers general algorithmic properties. I haven't checked it, but one paper the author mentioned was (dealing with complexity, too)
https://arxiv.org/abs/1106.5736

The Rubik's Cube is perhaps the world's most famous and iconic puzzle, well-known to have a rich underlying mathematical structure (group theory). In this paper, we show that the Rubik's Cube also has a rich underlying algorithmic structure. Specifically, we show that the n x n x n Rubik's Cube, as well as the n x n x 1 variant, has a "God's Number" (diameter of the configuration space) of Theta(n^2/log n). The upper bound comes from effectively parallelizing standard Theta(n^2) solution algorithms, while the lower bound follows from a counting argument. The upper bound gives an asymptotically optimal algorithm for solving a general Rubik's Cube in the worst case. Given a specific starting state, we show how to find the shortest solution in an n x O(1) x O(1) Rubik's Cube. Finally, we show that finding this optimal solution becomes NP-hard in an n x n x 1 Rubik's Cube when the positions and colors of some of the cubies are ignored (not used in determining whether the cube is solved).