#72340407

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1790
commutator	C₂×C₄⋊C₄	C₂³×C₄
Frattini, cosocle	C₂×#117	C₂⁴
prime radical	#7776879	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

4119 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(3,7)(5,8)(9,10)(11,19)(12,24)(13,27)(14,16)(15,22)(17,26)(18,20)(21,35)(23,33)(25,32)(28,30)(29,34)(31,38)(36,39)(37,40);(2,5,6,8)(9,11,20,22,14,23,13,28)(10,15,18,33,16,30,27,19)(12,21,25,38,26,37,29,36)(17,31,34,40,24,39,32,35);(10,16)(15,30)(17,24)(18,27)(19,33)(31,39)(32,34)(35,40);(1,2)(3,8)(4,6)(5,7)(11,21)(15,31)(19,35)(22,38)(23,37)(28,36)(30,39)(33,40)

Polycyclic presentation:

2^10|b^2=fg,e^2=hi,f^2=j,b^a=bf,c^a=cj,d^b=de,e^a=eh,e^d=ehi,f^a=fj,f^b=fj,f^d=fhj,f^e=fj,g^a=gj,g^b=gj,g^d=gij,g^e=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340415
maximal normal subgroup of	#72340415

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, 2, F.6*F.7);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);