#72225550

order	8192
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225549
commutator	C₈×C₃₂	C₂³×C₄
Frattini, cosocle	C₈×M₆₄	C₂⁴
prime radical	#72225451	C₂
LCS	G ▷ C₈×C₃₂ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₈×#346 ◁ #72216231 ◁ G
LECS	G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2371 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:

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

Polycyclic presentation:

2^13|a^2=m,c^2=f,d^2=gi,e^2=hj,g^2=ik,h^2=jl,i^2=k,j^2=lm,l^2=m,b^a=be,c^a=cl,c^b=cik,d^a=dh,d^b=dg,d^c=dk,e^a=eh,e^b=eh,e^c=elm,f^a=fm,f^b=fk,f^e=fm,g^a=gj,g^b=gi,h^a=hj,h^b=hj,h^c=hm,i^a=il,i^b=ik,j^a=jl,j^b=jl,k^a=km,l^a=lm,l^b=lm

Conjugacy classes & character table

224 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, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, 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, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.12);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
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.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.12*F.13);;
gap> SetConjugate(coll, 6, 1, F.6*F.13);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, 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, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.13);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> SetConjugate(coll, 9, 2, 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, 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> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);