#72264600

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23960
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3058	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2171 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)(6,8)(10,16,20,21)(12,26)(17,29,28,31)(18,27)(19,34,36,30)(23,37)(32,40,35,39)(33,38);(1,2)(3,6)(4,5)(7,8)(10,17,20,28)(11,22)(12,27)(13,14)(16,29,21,31)(18,26)(19,35,36,32)(23,33)(30,40,34,39)(37,38);(2,5)(3,7)(9,10,18,29,13,16,26,17,15,20,27,31,14,21,12,28)(11,19,33,39,25,30,37,32,22,36,38,40,24,34,23,35);(1,3,4,7)(2,6,5,8)(9,11,15,22)(10,19,20,36)(12,23,26,37)(13,25,14,24)(16,30,21,34)(17,32,28,35)(18,33,27,38)(29,39,31,40)

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=egh,d^2=hij,e^2=i,f^2=j,h^2=j,i^2=j,c^a=cf,c^b=ce,d^a=dghi,d^b=dj,d^c=dhi,e^a=ej,e^b=ei,e^c=ej,f^b=fj,f^d=fj,g^c=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264641
maximal normal subgroup of	#72264641

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