#72226979

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72226978
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	#33981	C₂×C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #129051 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1189 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:

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

Polycyclic presentation:

2^11|a^2=k,c^2=e,d^2=g,e^2=h,f^2=ij,i^2=jk,j^2=k,b^a=bf,c^a=ci,c^b=cfijk,d^a=djk,d^b=djk,d^c=dg,e^a=ek,e^b=ej,f^a=fi,f^b=fi,f^c=fijk,f^e=fk,g^a=gk,g^b=gk,h^b=hk,i^a=ij,i^b=ij,i^c=ijk,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226993, #72227138, #72227158
maximal normal subgroup of	#72226993, #72227138, #72227158

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