#72227093

order	8192
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227092
commutator, Frattini, cosocle	C₁₆×C₃₂	C₂⁴
LCS	G ▷ C₁₆×C₃₂ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ C₁₆×M₆₄ ◁ G
LECS	G ▷ C₁₆×C₃₂ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,9)(3,8,18,31)(4,10,24,33)(5,12,26,35)(6,14,25,16)(11,41,34,17)(13,45,36,19)(15,47,37,22)(20,27,51,58)(21,29,52,32)(23,38,53,40)(28,60,61,49)(30,63,59,48)(39,57,46,54)(42,56,44,50)(43,64,62,55);(3,17)(4,19)(5,22)(6,25)(8,11)(10,13)(12,15)(14,16)(18,41)(20,55)(21,56)(23,57)(24,45)(26,47)(27,43)(28,30)(29,44)(31,34)(32,42)(33,36)(35,37)(38,46)(39,40)(48,49)(50,52)(51,64)(53,54)(58,62)(59,61)(60,63);(2,8)(3,17)(4,19)(5,22)(6,25)(9,31)(10,28)(11,13)(12,30)(14,32)(15,27)(16,29)(18,41)(20,55)(21,56)(23,57)(24,45)(26,47)(33,61)(34,36)(35,59)(37,58)(38,62)(39,44)(40,43)(42,46)(48,49)(50,52)(51,64)(53,54)(60,63);(2,9)(3,18)(10,33)(12,35)(13,36)(14,16)(15,37)(17,41)(20,51)(21,52)(38,40)(39,46)(48,63)(49,60)(50,56)(55,64)

Polycyclic presentation:

2^13|a^2=m,e^2=gi,f^2=hj,g^2=ik,h^2=jl,i^2=k,j^2=lm,l^2=m,c^a=cf,c^b=ce,d^a=dm,d^c=dk,e^a=eh,e^b=eg,e^c=eg,f^a=fh,f^b=fh,f^c=fh,f^d=fm,g^a=gj,g^b=gi,g^c=gi,h^a=hj,h^b=hj,h^c=hj,i^a=il,i^b=ik,i^c=ik,j^a=jl,j^b=jl,j^c=jl,k^a=km,l^a=lm,l^b=lm,l^c=lm

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227170
maximal normal subgroup of	#72227170

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, 1, F.13);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetPower(coll, 8, F.10*F.12);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12*F.13);;
gap> SetPower(coll, 12, F.13);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.13);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.13);;
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);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
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);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> SetConjugate(coll, 10, 3, F.10*F.12);;
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);