#72318176

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	#7743196	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#3498 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

5203 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=hi,e^2=ij,f^2=ij,g^2=j,b^a=bg,c^a=chij,c^b=ce,d^a=dij,d^b=dghij,d^c=dij,e^a=ei,e^b=eh,e^c=ei,e^d=eij,f^a=fij,f^c=fhij,f^d=fi,f^e=fj,g^a=gj,g^b=gi,g^c=gij,g^d=gj,h^b=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318183, #72318229, #72318804
maximal normal subgroup of	#72318183, #72318229, #72318804

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