Most Niemeier matrices are characterized by their number of "roots" (vectors of norm 2). In this case we're lucky: we find (e.g. using the qfminim function in gp) that there are $528$ roots, and thus that each of the simple root lattices contained in the lattice has Coxeter number $528/24 = 22$; and this determines the lattice uniquely: it is the one with root lattice $D_{12}^2$.
In some cases there are two Niemeier lattices with the same Coxeter number, but they can be distinguished by the index of the sublattice generated by the roots (which can be obtained in gp with the command matdet(matrixqz(qfminim(G)[3],-1)) assuming G is the name of the Gram matrix).
To partly answer your earlier question: The Nebe-Sloane "Catalogue of Lattices" gives rational bases for all the Niemeier lattices (start from the second item in the $ D=24$ page), though the coordinates are given as floating-point real numbers so they might not be as useful as you might like.