#72225291

order	8192
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72214796
commutator	C₈×C₁₆	C₂³×C₈
Frattini, cosocle	C₈×#346	C₂⁴
prime radical	C₂×#72214187	C₄
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #72216161 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

6294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^13|b^2=e,d^2=eh,e^2=h,f^2=i,g^2=j,i^2=k,j^2=l,l^2=m,c^a=cg,c^b=cf,d^c=dik,e^c=ek,f^a=fj,f^b=fi,f^c=fik,f^d=fk,g^a=gjlm,g^b=gj,g^c=gjlm,g^d=gl,g^e=gm,i^a=il,i^b=ik,i^c=ik,j^a=jlm,j^b=jl,j^c=jlm,j^d=jm,k^a=km,l^a=lm,l^b=lm,l^c=lm

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

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", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 4, F.5*F.8);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetPower(coll, 12, F.13);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10*F.12*F.13);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10*F.12*F.13);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.13);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.12*F.13);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> SetConjugate(coll, 10, 3, F.10*F.12*F.13);;
gap> SetConjugate(coll, 10, 4, F.10*F.13);;
gap> SetConjugate(coll, 11, 1, F.11*F.13);;
gap> SetConjugate(coll, 12, 1, F.12*F.13);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> SetConjugate(coll, 12, 3, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);