#72227243

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#231631
commutator	C₈²	C₂³×C₈
Frattini, cosocle	C₄×C₈²	C₂⁴
prime radical	C₂×#123403	C₄
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₈² ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

2905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^12|a^2=e,c^2=fgjk,d^2=gi,e^2=h,f^2=ijkl,g^2=jk,h^2=l,i^2=k,j^2=l,b^a=bf,c^a=cfgjkl,c^b=cfgijl,d^a=dijk,d^b=dgijl,e^b=ej,e^c=ek,e^d=el,f^a=fi,f^b=fij,g^a=gl,g^b=gjkl,h^b=hl,i^a=ik,i^b=ik,j^b=jl

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72227244

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.6*F.7*F.10*F.11);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.10*F.11*F.12);;
gap> SetPower(coll, 7, F.10*F.11);;
gap> SetPower(coll, 8, F.12);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.10*F.11*F.12);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.9*F.10*F.12);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9*F.10*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);