#72225268

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72216176
commutator	C₈×C₁₆	C₂³×C₈
Frattini, cosocle	C₈×#346	C₂⁴
prime radical	#72216278	C₄
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ #72216186 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1024

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

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

Polycyclic presentation:

2^13|a^2=e,c^2=fi,d^2=gj,e^2=h,f^2=ik,g^2=jl,h^2=l,i^2=km,j^2=l,k^2=m,c^a=cik,c^b=cf,d^a=djl,d^b=dg,e^c=ek,e^d=el,f^a=fkm,f^b=fi,f^e=fm,g^a=gl,g^b=gj,h^c=hm,i^a=im,i^b=ik,j^b=jl,k^b=km

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225329
maximal normal subgroup of	#72225329

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