#72341516

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=i,d^2=fij,e^2=j,g^2=j,b^a=bhj,c^a=cij,c^b=cj,d^a=df,d^b=dg,d^c=dfgi,e^a=ehj,e^b=eij,e^c=ehi,e^d=eh,f^a=fj,f^b=fi,f^c=fij,f^d=fj,f^e=fj,g^a=gi,g^c=gij,g^d=gij,g^e=gj,g^f=gj,h^d=hj,i^d=ij

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341530
maximal normal subgroup of	#72341530

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