Classification of Finite p-Groups by the Size of Their Schur Multipliers

Let d(G) be the minimum number of elements required to generate a group G. For a group G of order pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^n$$\end{document} with a derived subgroup of order pk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ p^k $$\end{document} and d(G)=d,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d(G) = d,$$\end{document} we know the order of the Schur multiplier of G is bounded by p12(d-1)(n-k+2)+1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ p^{\frac{1}{2}(d-1)(n-k+2)+1}. $$\end{document} In this paper, we find the structure of all p-groups that obtains the mentioned bound. Moreover, we show that all of them are capable.