Character degrees of 5-groups of maximal class

Let 𝐺 be a 5-group of maximal class with major centralizer <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>/</m:mo> <m:msub> <m:mi>G</m:mi> <m:mn>4</m:mn> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> G₁=CG({G₂}/{G₄}) . In this paper, we prove that the irreducible character degrees of a 5-group 𝐺 of maximal class are almost determined by the irreducible character degrees of the major centralizer <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> G₁ and show that the set of irreducible character degrees of a 5-group of maximal class is either <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>5</m:mn> <m:mo>,</m:mo> <m:msup> <m:mn>5</m:mn> <m:mn>3</m:mn> </m:msup> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \{1,5,5³\} or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>5</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msup> <m:mn>5</m:mn> <m:mi>k</m:mi> </m:msup> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \{1,5,…,5ᵏ\} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> k≥ 1 .