By a Theorem of R. Guralnick and A. Lucchini (see MR1015993 and MR 1023965),(which does require the classification of finite simpe simple groups) the minimum number of generators for a finite group $G$ can exceed by at most one the maximum (over it all its Sylow subgroup subgroups $P$ of P$) f the minimum number of generators of$P$. It follows that the value of$D(G)$is between$d(H)$and$d(H)+1$for some$p$-subgroup$H$of$G.$Thus for a finite simple group$S,$the question does essentially come down to bounding the minimum number of generators of subgroups of$S$of prime power order, as was suggested might be the case in some comments. Th The sectional$p$-rank of a finite group$G$is the maximum number of generators of any section of$G$which is a$p$-group (a section of$G$is a group of the form$H/K$where$H$is a subgroup of$G$an$K \lhd H ).$Hence if we define the sectional rank of$G$to be the maximum of the minimum number of generators of an Abelian section of$G$, and denote it by$ar(G),$then we see that for any finite group$G$, simple or not, we have$ ar(G) \leq D(G) \leq ar(G)+1.$2 corrected typos, expanded explanation By a Theorem of R. Guralnick and A. Lucchini (see MR1015993 and MR 1023965),(which does require the classification of finite simpe groups) the minimum number of generators for a finite group$G$can exceed by at most one the maximum over it Sylow subgroup$P$of the minimum number of generators of$P$. It follows that the value of$D(G)$is between$d(H)$and$d(H)+1$for some$p$-subgroup$H$of$G.$Thus for a finite simple group$S,$the question does essentially to bounding the minimm minimum number of generators of subgroups of$S$of prime power order, as suggested might be the case in some comments. Th sectional$p$-rank of a finite group$G$is the maximum number of generators of any section of$G$which is a$p$-group (a section of$G$is a group of the form$H/K$where$H$is a subgroup of$G$an$K \lhd H ).$Hence if we define the sectional rank of$G$to be the maximum of the minimum number of generators of an Abelian section of$G$, and denote it by$ar(G),$then we see that for any finite group$G$, simple or not, we have$ ar(G) \leq D(G) \leq ar(G)+1.$1 By a Theorem of R. Guralnick and A. Lucchini (see MR1015993 and MR 1023965),(which does require the classification of finite simpe groups) the minimum number of generators for a finite group$G$can exceed by at most one the maximum over it Sylow subgroup$P$of the minimum number of generators of$P$. It follows that the value of$D(G)$is between$d(H)$and$d(H)+1$for some$p$-subgroup$H$of$G.$Thus for a finite simple group$S,$the question does essentially to bounding the minimm number of generators of subgroups of$S\$ of prime power order, as suggested might be the case in some comments.