#72309939

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#22670
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂×#509	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8282 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=fhj,c^2=ej,d^2=gj,f^2=h,g^2=i,i^2=j,b^a=bf,c^a=cf,c^b=cfh,d^a=dg,d^b=dhj,d^c=dhi,e^d=ej,f^a=fh,f^c=fh,g^a=gi,g^c=gj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309941, #72310092
maximal normal subgroup of	#72309941, #72310092

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