Small group number 9 of order 625

G is the group 625gp9

G has 2 minimal generators, rank 2 and exponent 25. The centre has rank 1.

The 6 maximal subgroups are: Ab(25,5), E125, M125 (4x).

There are 2 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 2, 2.

This cohomology ring calculation is complete.

Ring structure

The cohomology ring has 23 generators:

• y₁ in degree 1, a nilpotent element
• y₂ in degree 1, a nilpotent element
• x₁ in degree 2, a nilpotent element
• x₂ in degree 2, a nilpotent element
• x₃ in degree 2
• w in degree 3, a nilpotent element
• v₁ in degree 4, a nilpotent element
• v₂ in degree 4, a nilpotent element
• u in degree 5, a nilpotent element
• t₁ in degree 6, a nilpotent element
• t₂ in degree 6, a nilpotent element
• s in degree 7, a nilpotent element
• r₁ in degree 8, a nilpotent element
• r₂ in degree 8
• q₁ in degree 9, a nilpotent element
• q₂ in degree 9, a nilpotent element
• p₁ in degree 10
• p₂ in degree 10, a regular element
• o in degree 11, a nilpotent element
• n in degree 12
• m in degree 13, a nilpotent element
• l in degree 14
• k in degree 15, a nilpotent element

There are 230 minimal relations:

• y₂² = 0
• y₁.y₂ = 0
• y₁² = 0
• y₂.x₃ = 0
• y₂.x₂ = - 2y₁.x₂
• y₂.x₁ = y₁.x₂
• y₁.x₁ = 0
• x₂.x₃ = 0
• x₁.x₃ = 0
• x₂² = 0
• x₁.x₂ = 0
• x₁² = 0
• y₂.w = 0
• y₁.w = 0
• x₃.w = 0
• x₂.w = 0
• x₁.w = 0
• y₂.v₁ = 0
• y₁.v₂ = 0
• y₁.v₁ = 0
• x₃.v₂ = 0
• x₃.v₁ = 0
• w² = 0
• x₂.v₂ = 0
• x₂.v₁ = 0
• x₁.v₂ = 0
• x₁.v₁ = 0
• y₂.u = 0
• y₁.u = 0
• x₃.u = 0
• w.v₂ = 0
• w.v₁ = 0
• x₂.u = 0
• x₁.u = 0
• y₂.t₁ = 0
• y₁.t₂ = 0
• y₁.t₁ = 0
• x₃.t₂ = 0
• x₃.t₁ = 0
• v₂² = 0
• v₁.v₂ = 0
• v₁² = 0
• w.u = 0
• x₂.t₂ = 0
• x₂.t₁ = 0
• x₁.t₂ = 0
• x₁.t₁ = 0
• y₂.s = 0
• y₁.s = 0
• x₃.s = 0
• y₁.r₂ = 0
• v₂.u = 0
• v₁.u = 0
• w.t₂ = 0
• w.t₁ = 0
• x₂.s = 0
• x₁.s = 0
• y₂.r₁ = 0
• y₁.r₁ = 0
• x₃.r₂ = 2y₁.q₂
• x₃.r₁ = - y₁.q₂
• x₂.r₂ = y₂.q₁
• x₁.r₂ = 0
• u² = 0
• v₂.t₂ = 0
• v₂.t₁ = 0
• v₁.t₂ = 0
• v₁.t₁ = 0
• w.s = 0
• x₂.r₁ = 0
• x₁.r₁ = 0
• y₂.q₂ = 2y₂.q₁
• y₁.q₁ = 0
• w.r₂ = y₂.p₁
• x₃.q₁ = 0
• y₁.p₁ = 0
• u.t₂ = 0
• u.t₁ = 0
• v₂.s = 0
• v₁.s = 0
• w.r₁ = 0
• x₂.q₂ = 0
• x₂.q₁ = 0
• x₁.q₂ = 0
• x₁.q₁ = 0
• x₃.p₁ = - 2y₁.x₃.q₂
• v₂.r₂ = 2y₂.o
• v₁.r₂ = y₂.o
• x₂.p₁ = y₂.o
• x₁.p₁ = 0
• t₂² = 0
• t₁.t₂ = 0
• t₁² = 0
• u.s = 0
• v₂.r₁ = 0
• v₁.r₁ = 0
• w.q₂ = 2y₂.o
• w.q₁ = y₂.o
• y₁.o = - 2y₁.x₃.q₂
• u.r₂ = y₂.n - y₁.x₂.p₂
• w.p₁ = y₂.n - y₁.x₂.p₂
• x₃.o = - 2x₃².q₂ + y₁.x₃.p₂
• y₁.n = - y₁.x₃.p₂ - 2y₁.x₂.p₂
• t₂.s = 0
• t₁.s = 0
• u.r₁ = 0
• v₂.q₂ = 0
• v₂.q₁ = 0
• v₁.q₂ = 0
• v₁.q₁ = 0
• x₂.o = y₁.x₂.p₂
• x₁.o = 0
• x₃.n = - x₃².p₂ + y₁.x₃².q₂
• t₂.r₂ = - 2y₂.m
• t₁.r₂ = 2y₂.m
• v₂.p₁ = 2y₂.m
• v₁.p₁ = y₂.m
• x₂.n = y₂.m
• x₁.n = 0
• s² = 0
• t₂.r₁ = 0
• t₁.r₁ = 0
• u.q₂ = 2y₂.m
• u.q₁ = y₂.m
• w.o = y₂.m
• y₁.m = 2y₁.x₃².q₂
• s.r₂ = 2y₂.l
• u.p₁ = y₂.l
• w.n = y₂.l
• x₃.m = 2x₃³.q₂
• y₁.l = y₁.x₃².p₂
• s.r₁ = 0
• t₂.q₂ = 0
• t₂.q₁ = 0
• t₁.q₂ = 0
• t₁.q₁ = 0
• v₂.o = 0
• v₁.o = 0
• x₂.m = 0
• x₁.m = 0
• x₃.l = x₃³.p₂ + 2y₁.x₃³.q₂
• r₁.r₂ = - 2y₂.k
• t₂.p₁ = - 2y₂.k
• t₁.p₁ = 2y₂.k
• v₂.n = 2y₂.k
• v₁.n = y₂.k
• x₂.l = y₂.k
• x₁.l = 0
• r₁² = 0
• s.q₂ = - y₂.k
• s.q₁ = 2y₂.k
• u.o = y₂.k
• w.m = y₂.k
• y₁.k = - y₁.x₃³.q₂
• r₂.q₂ = 2r₂.q₁ + 2y₂.r₂²
• s.p₁ = y₂.r₂²
• u.n = - 2y₂.r₂²
• w.l = - 2y₂.r₂²
• x₃.k = - x₃⁴.q₂
• r₁.q₂ = 0
• r₁.q₁ = 0
• t₂.o = 0
• t₁.o = 0
• v₂.m = 0
• v₁.m = 0
• x₂.k = 0
• x₁.k = 0
• r₁.p₁ = - y₂.r₂.q₁
• t₂.n = - y₂.r₂.q₁
• t₁.n = y₂.r₂.q₁
• v₂.l = y₂.r₂.q₁
• v₁.l = - 2y₂.r₂.q₁
• q₂² = 0
• q₁.q₂ = - 2y₂.r₂.q₁
• q₁² = 0
• s.o = y₂.r₂.q₁
• u.m = - 2y₂.r₂.q₁
• w.k = - 2y₂.r₂.q₁
• q₂.p₁ = 2r₂.o - y₂.r₂.p₁
• q₁.p₁ = r₂.o + y₂.r₂.p₁
• s.n = y₂.r₂.p₁
• u.l = - 2y₂.r₂.p₁
• r₁.o = 0
• t₂.m = 0
• t₁.m = 0
• v₂.k = 0
• v₁.k = 0
• p₁² = r₂.n + y₂.r₂.o + 2y₂.q₁.p₂ + 2y₁.q₂.p₂
• r₁.n = - y₂.r₂.o + y₁.q₂.p₂
• t₂.l = - y₂.r₂.o
• t₁.l = y₂.r₂.o
• q₂.o = - y₂.r₂.o - y₁.q₂.p₂
• q₁.o = y₂.r₂.o
• s.m = y₂.r₂.o
• u.k = - 2y₂.r₂.o
• p₁.o = r₂.m - 2y₂.p₁.p₂
• q₂.n = 2r₂.m - y₂.r₂.n - x₃.q₂.p₂ + y₂.p₁.p₂
• q₁.n = r₂.m + y₂.r₂.n - 2y₂.p₁.p₂
• s.l = y₂.r₂.n
• r₁.m = 0
• t₂.k = 0
• t₁.k = 0
• p₁.n = r₂.l + 2y₂.r₂.m + y₂.p₂.o
• r₁.l = - y₂.r₂.m - y₁.x₃.q₂.p₂
• o² = 0
• q₂.m = - y₂.r₂.m + y₂.p₂.o
• q₁.m = y₂.r₂.m - 2y₂.p₂.o
• s.k = y₂.r₂.m
• o.n = r₂.k + 2x₃².q₂.p₂ + y₂.p₂.n - y₁.x₃.p₂² + 2y₁.x₂.p₂²
• p₁.m = r₂.k - 2y₂.p₂.n + 2y₁.x₂.p₂²
• q₂.l = 2r₂.k - y₂.r₂.l + x₃².q₂.p₂ + 2y₂.p₂.n - 2y₁.x₂.p₂²
• q₁.l = r₂.k + y₂.r₂.l + y₂.p₂.n - y₁.x₂.p₂²
• r₁.k = 0
• n² = - 2r₂³ + x₃².p₂² - y₂.r₂.k + y₂.p₂.m - 2y₁.x₃².q₂.p₂
• p₁.l = - 2r₂³ - 2y₂.r₂.k + 2y₂.p₂.m - 2y₁.x₃².q₂.p₂
• o.m = - 2y₂.p₂.m + 2y₁.x₃².q₂.p₂
• q₂.k = - y₂.r₂.k + 2y₂.p₂.m
• q₁.k = y₂.r₂.k + y₂.p₂.m
• n.m = - 2r₂².q₁ + 2y₂.r₂³ - 2x₃³.q₂.p₂ + 2y₂.p₂.l
• o.l = - 2r₂².q₁ + 2y₂.r₂³ - 2x₃³.q₂.p₂ + y₁.x₃².p₂²
• p₁.k = - 2r₂².q₁ + 2y₂.r₂³ - y₂.p₂.l
• n.l = - 2r₂².p₁ - x₃³.p₂² + y₂.r₂².q₁ - y₁.x₃³.q₂.p₂
• m² = 0
• o.k = y₂.p₂.k - y₁.x₃³.q₂.p₂
• m.l = - 2r₂².o + 2x₃⁴.q₂.p₂ + y₂.r₂².p₂
• n.k = - 2r₂².o + x₃⁴.q₂.p₂ + 2y₂.r₂².p₂
• l² = - 2r₂².n + x₃⁴.p₂² - 2y₂.r₂².o - 2y₂.r₂.q₁.p₂ - y₁.x₃⁴.q₂.p₂
• m.k = - y₂.r₂.q₁.p₂
• l.k = - 2r₂².m - x₃⁵.q₂.p₂ + y₂.r₂.p₁.p₂
• k² = 0

This minimal generating set constitutes a Gröbner basis for the relations ideal.

Essential ideal: There are 3 minimal generators:

• y₁.x₂
• y₂.v₂
• y₂.t₂

Nilradical: There are 17 minimal generators:

• y₂
• y₁
• x₂
• x₁
• w
• v₂
• v₁
• u
• t₂
• t₁
• s
• r₁
• q₂
• q₁
• o
• m
• k

Completion information

This cohomology ring was obtained from a calculation out to degree 30. The cohomology ring approximation is stable from degree 30 onwards, and Carlson's tests detect stability from degree 30 onwards.

This cohomology ring has dimension 2 and depth 1. Here is a homogeneous system of parameters:

• h₁ = p₂ in degree 10
• h₂ = r₂ + x₃⁴ in degree 8

The first term h₁ forms a regular sequence of maximum length. The remaining term h₂ is annihilated by the class x₁.

The first term h₁ forms a complete Duflot regular sequence. That is, its restriction to the greatest central elementary abelian subgroup forms a regular sequence of maximal length.

The ideal of essential classes is free of rank 3 as a module over the polynomial algebra on h₁. These free generators are:

• G₁ = y₁.x₂ in degree 3
• G₂ = y₂.v₂ in degree 5
• G₃ = y₂.t₂ in degree 7

Koszul information

A basis for R/(h₁, h₂) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 18.

• 1 in degree 0
• y₂ in degree 1
• y₁ in degree 1
• x₃ in degree 2
• x₂ in degree 2
• x₁ in degree 2
• w in degree 3
• y₁.x₃ in degree 3
• y₁.x₂ in degree 3
• x₃² in degree 4
• v₂ in degree 4
• v₁ in degree 4
• u in degree 5
• y₁.x₃² in degree 5
• y₂.v₂ in degree 5
• x₃³ in degree 6
• t₂ in degree 6
• t₁ in degree 6
• s in degree 7
• y₁.x₃³ in degree 7
• y₂.t₂ in degree 7
• x₃⁴ in degree 8
• r₁ in degree 8
• q₂ in degree 9
• q₁ in degree 9
• p₁ in degree 10
• y₁.q₂ in degree 10
• o in degree 11
• x₃.q₂ in degree 11
• n in degree 12
• y₁.x₃.q₂ in degree 12
• m in degree 13
• x₃².q₂ in degree 13
• l in degree 14
• y₁.x₃².q₂ in degree 14
• k in degree 15
• x₃³.q₂ in degree 15
• y₁.x₃³.q₂ in degree 16

A basis for Ann_R/(h1)(h₂) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 10.

• x₁ in degree 2
• y₁.x₂ in degree 3
• v₂ - 2v₁ in degree 4
• y₂.v₂ - 2y₂.v₁ in degree 5
• t₂ + t₁ in degree 6
• y₂.t₂ + y₂.t₁ in degree 7

Restriction information

• Restrictions to maximal subgroups
• Restrictions to maximal elementary abelian subgroups
• Restriction to the greatest central elementary abelian subgroup

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is 125gp2

• y₁ restricts to 0
• y₂ restricts to y₁
• x₁ restricts to 0
• x₂ restricts to y₁.y₂
• x₃ restricts to 0
• w restricts to y₁.x₁
• v₁ restricts to y₁.y₂.x₁
• v₂ restricts to 2y₁.y₂.x₁
• u restricts to y₁.x₁²
• t₁ restricts to 2y₁.y₂.x₁²
• t₂ restricts to - 2y₁.y₂.x₁²
• s restricts to 2y₁.x₁³
• r₁ restricts to - 2y₁.y₂.x₁³
• r₂ restricts to 2x₁⁴ - y₁.y₂.x₁³
• q₁ restricts to 2y₂.x₁⁴ - 2y₁.x₁³.x₂
• q₂ restricts to - y₂.x₁⁴ + y₁.x₁³.x₂ - y₁.x₁⁴
• p₁ restricts to 2x₁⁵ - 2y₁.y₂.x₁⁴
• p₂ restricts to - 2x₂⁵ + 2x₁⁴.x₂ + x₁⁵ + y₁.y₂.x₁⁴
• o restricts to 2y₂.x₁⁵ - 2y₁.x₁⁴.x₂ - 2y₁.x₁⁵
• n restricts to 2x₁⁶ - y₁.y₂.x₂⁵ + y₁.y₂.x₁⁴.x₂ - 2y₁.y₂.x₁⁵
• m restricts to 2y₂.x₁⁶ + y₁.x₁.x₂⁵ + 2y₁.x₁⁵.x₂
• l restricts to 2x₁⁷ + y₁.y₂.x₁.x₂⁵ - y₁.y₂.x₁⁵.x₂ + 2y₁.y₂.x₁⁶
• k restricts to 2y₂.x₁⁷ + 2y₁.x₁².x₂⁵ + y₁.x₁⁶.x₂ + 2y₁.x₁⁷

Restriction to maximal subgroup number 2, which is 125gp3

• y₁ restricts to y₂ + 2y₁
• y₂ restricts to 0
• x₁ restricts to 0
• x₂ restricts to 0
• x₃ restricts to 2x₄ + x₃
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to y₂.w₁ - y₁.w₂
• u restricts to 0
• t₁ restricts to 0
• t₂ restricts to w₁.w₂ - 2y₁.x₃.w₂ - 2y₁.x₃.w₁
• s restricts to 0
• r₁ restricts to - 2y₂.x₃².w₂ - 2y₁.x₄².w₁ + y₁.x₃.x₄.w₁ - y₁.x₃².w₂ + 2y₁.x₃².w₁
• r₂ restricts to - 2r + 2x₃.x₄³ + x₃³.x₄ - 2y₂.x₃².w₂ - 2y₁.x₄².w₁ - y₁.x₃.x₄.w₁ - 2y₁.x₃².w₁
• q₁ restricts to - y₁.x₃.x₄³ - 2y₁.x₃².x₄² + y₁.x₃³.x₄ + 2y₁.x₃⁴
• q₂ restricts to 2x₄³.w₁ - x₃.x₄².w₁ - 2x₃².x₄.w₁ + 2x₃³.w₂ + x₃³.w₁ - y₁.x₃.x₄³ - 2y₁.x₃².x₄² + y₁.x₃³.x₄ + y₁.x₃⁴
• p₁ restricts to - x₃.x₄⁴ - 2x₃².x₄³ + x₃³.x₄² + 2x₃⁴.x₄ + y₂.x₃³.w₂ + 2y₁.x₄³.w₁
• p₂ restricts to x₃.x₄⁴ - 2x₃².x₄³ - x₃³.x₄² - x₃⁴.x₄ - 2p - 2y₂.x₃³.w₂ + 2y₁.x₄³.w₁ + y₁.x₃.x₄².w₁ - y₁.x₃².x₄.w₁ - y₁.x₃³.w₂ + y₁.x₃³.w₁
• o restricts to 2x₄⁴.w₁ + x₃⁴.w₂ - y₁.x₃².x₄³ - 2y₂.p + y₁.p
• n restricts to - 2x₃³.x₄³ + x₃⁴.x₄² - x₄.p + 2x₃.p - y₂.x₃⁴.w₂ - y₁.x₄⁴.w₁ + y₁.x₃².x₄².w₁ - y₁.x₃³.x₄.w₁ - y₁.x₃⁴.w₂ + y₁.x₃⁴.w₁
• m restricts to x₄⁵.w₁ - x₃⁵.w₂ + x₃⁵.w₁ + y₁.x₃³.x₄³ + 2y₁.x₃⁴.x₄² + 2y₁.x₃⁵.x₄ - y₁.x₃⁶
• l restricts to 2x₃⁴.x₄³ - 2x₃⁵.x₄² - 2x₃⁶.x₄ + 2x₄².p + 2x₃.x₄.p - 2x₃².p + 2y₂.x₃⁵.w₂ + y₁.x₃³.x₄².w₁ - y₁.x₃⁴.x₄.w₁ + 2y₁.x₃⁵.w₂ - y₁.x₃⁵.w₁
• k restricts to - x₄⁶.w₁ + x₃⁵.x₄.w₁ - 2x₃⁶.w₂ - 2x₃⁶.w₁ - 2y₁.x₃⁴.x₄³ - 2y₁.x₃⁵.x₄² - y₁.x₃⁷

Restriction to maximal subgroup number 3, which is 125gp4

• y₁ restricts to - 2y₁
• y₂ restricts to 2y₁
• x₁ restricts to 2y₁.y₂
• x₂ restricts to - 2y₁.y₂
• x₃ restricts to 2y₁.y₂
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to 2y₂.w
• u restricts to 0
• t₁ restricts to 0
• t₂ restricts to - y₂.u
• s restricts to 0
• r₁ restricts to 0
• r₂ restricts to 2x⁴ + 2y₂.s
• q₁ restricts to 2y₂.x⁴
• q₂ restricts to - y₂.x⁴
• p₁ restricts to 2x⁵
• p₂ restricts to x⁵ - p + y₂.q
• o restricts to 2y₂.x⁵ + 2y₁.p
• n restricts to 2x⁶ - y₁.y₂.p
• m restricts to 2y₂.x⁶
• l restricts to 2x⁷
• k restricts to 2y₂.x⁷

Restriction to maximal subgroup number 4, which is 125gp4

• y₁ restricts to - 2y₁
• y₂ restricts to - y₁
• x₁ restricts to - y₁.y₂
• x₂ restricts to 0
• x₃ restricts to 2y₁.y₂
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to - 2y₂.w
• u restricts to 0
• t₁ restricts to 0
• t₂ restricts to 2y₂.u
• s restricts to 0
• r₁ restricts to 0
• r₂ restricts to 2x⁴ + 2y₂.s
• q₁ restricts to - y₂.x⁴
• q₂ restricts to - 2y₂.x⁴
• p₁ restricts to - x⁵
• p₂ restricts to 2x⁵ - 2p + 2y₂.q
• o restricts to - 2y₂.x⁵ - y₁.p
• n restricts to - 2x⁶ - 2y₁.y₂.p
• m restricts to y₂.x⁶
• l restricts to x⁷
• k restricts to 2y₂.x⁷

Restriction to maximal subgroup number 5, which is 125gp4

• y₁ restricts to - 2y₁
• y₂ restricts to - 2y₁
• x₁ restricts to - 2y₁.y₂
• x₂ restricts to y₁.y₂
• x₃ restricts to 2y₁.y₂
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to 2y₂.w
• u restricts to 0
• t₁ restricts to 0
• t₂ restricts to y₂.u
• s restricts to 0
• r₁ restricts to 0
• r₂ restricts to 2x⁴ + 2y₂.s
• q₁ restricts to - 2y₂.x⁴
• q₂ restricts to y₂.x⁴
• p₁ restricts to - 2x⁵
• p₂ restricts to - x⁵ + p - y₂.q
• o restricts to 2y₂.x⁵ - 2y₁.p
• n restricts to 2x⁶ + 2y₁.y₂.p
• m restricts to - 2y₂.x⁶
• l restricts to - 2x⁷
• k restricts to 2y₂.x⁷

Restriction to maximal subgroup number 6, which is 125gp4

• y₁ restricts to 2y₁
• y₂ restricts to - y₁
• x₁ restricts to - y₁.y₂
• x₂ restricts to - y₁.y₂
• x₃ restricts to - 2y₁.y₂
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to - 2y₂.w
• u restricts to 0
• t₁ restricts to 0
• t₂ restricts to 2y₂.u
• s restricts to 0
• r₁ restricts to 0
• r₂ restricts to 2x⁴ + 2y₂.s
• q₁ restricts to y₂.x⁴
• q₂ restricts to 2y₂.x⁴
• p₁ restricts to x⁵
• p₂ restricts to - 2x⁵ - 2p + 2y₂.q
• o restricts to - 2y₂.x⁵ + y₁.p
• n restricts to - 2x⁶ + y₁.y₂.p
• m restricts to - y₂.x⁶
• l restricts to - x⁷
• k restricts to 2y₂.x⁷

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V25

• y₁ restricts to y₂
• y₂ restricts to 0
• x₁ restricts to 0
• x₂ restricts to 0
• x₃ restricts to x₂
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to 0
• u restricts to 0
• t₁ restricts to 0
• t₂ restricts to 0
• s restricts to 0
• r₁ restricts to - 2y₁.y₂.x₂³
• r₂ restricts to - y₁.y₂.x₂³
• q₁ restricts to 0
• q₂ restricts to 2y₂.x₁.x₂³ - 2y₁.x₂⁴
• p₁ restricts to y₁.y₂.x₂⁴
• p₂ restricts to - 2x₁.x₂⁴ + 2x₁⁵ - y₁.y₂.x₂⁴
• o restricts to - y₂.x₁.x₂⁴ + 2y₂.x₁⁵ - y₁.x₂⁵
• n restricts to 2x₁.x₂⁵ - 2x₁⁵.x₂ - 2y₁.y₂.x₂⁵
• m restricts to - y₂.x₁.x₂⁵ + y₁.x₂⁶
• l restricts to - 2x₁.x₂⁶ + 2x₁⁵.x₂² - 2y₁.y₂.x₂⁶
• k restricts to - 2y₂.x₁.x₂⁶ + 2y₁.x₂⁷

Restriction to maximal elementary abelian number 2, which is V25

• y₁ restricts to 0
• y₂ restricts to 0
• x₁ restricts to 0
• x₂ restricts to 0
• x₃ restricts to 0
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to 0
• u restricts to 0
• t₁ restricts to 0
• t₂ restricts to 0
• s restricts to 0
• r₁ restricts to 0
• r₂ restricts to 2x₁⁴
• q₁ restricts to - 2y₁.x₁⁴
• q₂ restricts to y₁.x₁⁴
• p₁ restricts to - 2x₁⁵
• p₂ restricts to 2x₂⁵ - 2x₁⁴.x₂ - x₁⁵
• o restricts to 2y₁.x₁⁵
• n restricts to 2x₁⁶
• m restricts to - 2y₁.x₁⁶
• l restricts to - 2x₁⁷
• k restricts to 2y₁.x₁⁷

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is C5

• y₁ restricts to 0
• y₂ restricts to 0
• x₁ restricts to 0
• x₂ restricts to 0
• x₃ restricts to 0
• w restricts to 0
• v₁ restricts to 0
• v₂ restricts to 0
• u restricts to 0
• t₁ restricts to 0
• t₂ restricts to 0
• s restricts to 0
• r₁ restricts to 0
• r₂ restricts to 0
• q₁ restricts to 0
• q₂ restricts to 0
• p₁ restricts to 0
• p₂ restricts to - 2x⁵
• o restricts to 0
• n restricts to 0
• m restricts to 0
• l restricts to 0
• k restricts to 0

Poincaré series

(1 + 2t + 3t² + 3t³ + 3t⁴ + 3t⁵ + 3t⁶ + 3t⁷ + 2t⁸ + 2t⁹ + t¹⁰ + t¹¹ + t¹² + t¹³ + t¹⁴ + t¹⁵ + t¹⁶) / (1 - t⁸) (1 - t¹⁰)