#72226982

order	8192
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72226981
commutator	C₈×C₁₆	C₂⁴×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂⁵
prime radical	#72225593	C₂
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

9844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^13|c^2=f,d^2=gi,e^2=hj,g^2=ik,h^2=jl,i^2=k,j^2=lm,l^2=m,c^a=cl,c^b=ck,d^a=dh,d^b=dg,d^c=dk,e^a=eh,e^b=eh,e^c=elm,f^a=fm,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

284 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, 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, 3, 1, F.3*F.12);;
gap> SetConjugate(coll, 3, 2, F.3*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, 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);