#72341626

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	#7743196	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5513 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:

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

Polycyclic presentation:

2^10|a^2=e,d^2=ij,e^2=j,f^2=j,b^a=bg,c^a=cf,d^a=dgi,d^b=dij,e^b=ei,e^d=eij,f^a=fj,f^b=fh,f^c=fj,f^d=fh,g^a=gi,g^c=ghj,g^d=gj,g^e=gj,g^f=gj,h^a=hj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341642
maximal normal subgroup of	#72341642

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