The classical Hermite constant acounts for the highest density of a regular sphere packing. A generalised version of it consists more broadly in bounding the size of the biggest minimal sub-structure in some given object, such as the height of a rational flag (nested vector spaces) of fixed shape on some number field of given dimension. We shall show how one can obtain some values of this new Hermite constant, how to characterize the flags that locally achieve the constant in the spirit of Voronoi work and developp a technique based on representation theory and designs to exhibit such examples.