#72303568

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	C₂×C₄	C₂×B(4,𝔽₂)
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
socle	C₂²	C₂×#1790
prime radical	C₂³×#41	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

5135 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=hj,c^2=e,e^2=j,b^a=bhj,c^a=cf,c^b=cfg,d^a=dj,f^a=fj,f^b=fi,g^a=gij,g^b=gi,g^c=gj,g^f=gj,h^c=hj,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303581
maximal normal subgroup of	#72303581
maximal quotient of	#72303617

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