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E8 has dimension 248 (as a complex manifold). Its rank, which is the dimension of its maximal torus, is 8. Therefore the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article. The Weyl group of E8, which is the group of symmetries of the maximal torus which are induced by conjugations in the whole group, has order 696729600.
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E8 is unique among simple Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra E8 itself; it is also the unique one which has the following three properties: trivial center, simply connected, and simply laced (all roots have the same length).
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There is a Lie algebra En for every integer n ≥ 3, which is infinite dimensional if n is greater than 8.
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