#72341157

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32898
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4251 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:

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

Polycyclic presentation:

2^10|b^2=j,c^2=ij,g^2=j,b^a=be,c^a=cij,c^b=cf,d^a=dhij,d^b=dfgj,d^c=dhi,e^c=ej,e^d=eh,f^c=fj,f^d=fhi,g^a=ghj,g^c=ghij,g^d=ghi,g^f=gj,i^b=ij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341171
maximal normal subgroup of	#72341171

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