#72314381

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	#61234	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ D₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2478 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=f,d^2=fhj,e^2=fghi,g^2=i,i^2=j,c^b=ch,d^a=dhj,d^b=dj,d^c=dj,e^a=ej,e^b=eg,e^c=eij,e^d=egh,f^e=fj,g^b=gi,g^c=gj,g^d=gi,g^e=gj,i^b=ij,i^d=ij

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314408, #72314559, #72314619
maximal normal subgroup of	#72314408, #72314559, #72314619

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