#72318659

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	#20609
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#7826509	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5265 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)(3,6,7,8)(4,5)(9,10)(11,21,24,16)(12,25)(13,18,14,19)(15,20)(17,28,34,29)(22,37,26,38)(23,35)(27,30)(31,33)(32,39)(36,40);(2,5)(3,7)(9,11)(10,16)(12,26)(13,31)(14,23)(15,24)(17,32)(18,35)(19,33)(20,21)(22,30)(25,37)(27,38)(28,39)(29,40)(34,36);(9,12,15,30)(10,17)(11,22,24,26)(13,28,14,29)(16,32)(18,25)(19,27)(20,34)(21,36)(23,39,31,40)(33,38)(35,37);(1,3)(2,6,5,8)(4,7)(9,10)(11,21,24,16)(12,27)(13,18)(14,19)(15,20)(17,28)(22,38,26,37)(23,35,31,33)(25,30)(29,34)(32,39,36,40)

Polycyclic presentation:

2^10|a^2=f,c^2=h,d^2=eg,e^2=j,g^2=j,b^a=bij,c^a=ce,c^b=ch,d^a=defgj,d^b=dij,d^c=deh,e^a=eh,e^b=ej,f^b=fj,f^c=fhj,f^d=fhi,f^e=fj,g^a=gi,g^f=gj,h^a=hj,h^d=hj,i^a=ij,i^d=ij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318703
maximal normal subgroup of	#72318703

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