Schur multipliers of 𝑝-groups of maximal class

We give bounds for the order of the Schur multiplier of a group of maximal class and order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> pⁿ for all primes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> p>2 , and we give examples to show that our bounds are best possible. In particular, for every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> p>2 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> n≥ 3 , we construct groups where the orders of their Schur multipliers attain our bounds. We describe the Schur multipliers of the 3-groups of maximal class, and we compute the Schur multipliers of the groups of maximal class of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> pⁿ for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> p>2 and all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mn>7</m:mn> </m:mrow> </m:math> n≤ 7 .