#72227278

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#9618
commutator	C₈²	C₂²×C₈
Frattini, cosocle	C₄×C₈²	C₂³
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₈² ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 2048

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^11|a^2=f,b^2=d,c^2=dhik,d^2=g,e^2=hi,f^2=ijk,h^2=j,i^2=k,b^a=bfijk,c^a=ce,c^b=cfhij,d^a=dk,d^c=djk,e^a=eh,e^b=ei,e^c=ehi,f^b=fijk,f^c=fijk,h^a=hj,h^c=hj,i^b=ik,i^c=ik

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227290
maximal normal subgroup of	#72227290
maximal quotient of	#72227279

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4*F.8*F.9*F.11);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.9*F.10*F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);