Lie groups hierarchy in connection with the derivation of the inverse electromagnetic fine structure constant from the number of particle-like states 548, 576 and 672

In this work we present different mathematical derivations of the inverse electromagnetic fine structure constant from the dimension of the compactified Klein modular curve, the dimension of the exceptional Lie group E8 E8 and from the number of states 548, 576 and 672. We show the close connection of the number of particle-like states 548, 576 and 672 with the dimensions of Lie groups, specially with E71/2 and other exceptional Lie groups.