#72322205

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#31064
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#498795	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,12,17,13,18,10,15,9,8,26,27,29,28,11,16)(7,19,24,14,25,33,23,34,20,32,30,21,31,35,22,36);(1,3,4,2)(5,7,9,20)(6,14,8,21)(10,24,11,30)(12,23,26,22)(13,31,29,25)(15,33,16,35)(17,32,27,19)(18,36,28,34);(2,3)(6,8)(7,20)(14,21)(15,16)(17,27)(18,28)(19,32)(22,23)(24,30)(25,31)(33,35)(34,36);(2,3)(5,8,9,6)(7,21)(10,17,11,27)(12,15,26,16)(13,18,29,28)(14,20)(19,30)(22,35)(23,33)(24,32)(25,34)(31,36)

Polycyclic presentation:

2^10|a^2=h,b^2=fgi,d^2=ghj,f^2=ij,g^2=ij,h^2=i,i^2=j,b^a=bf,c^a=cfh,c^b=ce,d^a=dfi,d^b=de,d^c=dghj,e^a=ej,e^d=ej,f^b=fi,f^d=fi,g^b=gi,g^d=gi,h^b=hi,h^d=hij,i^b=ij,i^d=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322221, #72322428, #72323029
maximal normal subgroup of	#72322221, #72322428, #72323029

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 2, F.6*F.7*F.9);;
gap> SetPower(coll, 4, F.7*F.8*F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);