#72314164

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	#7622935	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ #20475 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2959 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=fhj,d^2=fh,e^2=ghj,g^2=i,i^2=j,b^a=bh,c^a=chj,d^a=dhj,d^c=dj,e^a=eh,e^b=eg,e^c=ei,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

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314251, #72314371, #72315244
maximal normal subgroup of	#72314251, #72314371, #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, 1, F.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.8*F.10);;
gap> SetPower(coll, 4, F.6*F.8);;
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, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*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);;
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);