#72314485

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7623725	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ #20495 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|b^2=f,c^2=f,d^2=fhj,e^2=ghj,g^2=i,i^2=j,c^a=ch,c^b=ch,d^a=dhj,d^b=dj,d^c=dj,e^a=eh,e^b=eg,e^c=eij,e^d=egh,f^e=fj,g^b=gi,g^c=gj,g^d=gi,i^b=ij,i^d=ij

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314497, #72315187, #72315244
maximal normal subgroup of	#72314497, #72315187, #72315244

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