#72340726

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1781
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂×#117	C₂⁴
prime radical	#7714980	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#535 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

5617 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=hi,e^2=hj,g^2=j,c^a=ce,c^b=cj,d^a=dfgj,d^c=dh,e^a=ei,e^c=eh,e^d=eh,f^c=fhij,f^e=fj,g^a=gj,g^c=gij,g^d=gj,g^e=gj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340729
maximal normal subgroup of	#72340729

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