#72341138

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	#7739587	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5771 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:

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

Polycyclic presentation:

2^10|b^2=f,g^2=j,b^a=be,c^b=ch,d^a=dij,d^b=dfgj,d^c=di,e^d=ei,f^c=fj,f^d=fh,g^a=gi,g^b=gh,g^d=gh,g^e=gj,g^f=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341171
maximal normal subgroup of	#72341171

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