#72341162

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#20585
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#8924809	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #35346 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4127 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,23)(13,27)(14,17)(16,25)(18,28)(19,29)(20,30)(21,36)(22,31)(24,32)(26,33)(34,39)(35,40)(37,38);(6,8)(9,11)(10,15)(12,24,25,19)(13,21,28,20)(14,22)(16,32,23,29)(17,31)(18,30,27,36)(26,35)(33,39)(34,38)(37,40);(2,5)(6,8)(9,12)(10,16)(11,19)(13,26)(14,25)(15,29)(17,23)(18,33)(20,35)(21,34)(22,24)(27,37)(28,38)(30,40)(31,32)(36,39);(1,3,4,7)(2,6,5,8)(10,17)(11,20)(12,25)(13,28)(15,30)(19,34)(21,22)(24,35)(29,39)(31,36)(32,40)(33,37)

Polycyclic presentation:

2^10|b^2=f,d^2=h,g^2=j,b^a=be,c^a=cij,c^b=chi,d^a=dhij,d^b=dg,d^c=dj,e^c=ej,e^d=eh,f^c=fj,f^d=fhi,g^a=ghj,g^b=ghij,g^d=gj,g^f=gj,i^b=ij

Conjugacy classes & character table

70 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.6);;
gap> SetPower(coll, 4, F.8);;
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.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
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, 2, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
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);