#72227151

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	#72227150
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	#512952	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ #128258 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1163 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:

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

Polycyclic presentation:

2^11|a^2=g,b^2=g,d^2=e,e^2=h,f^2=ij,g^2=k,i^2=jk,j^2=k,b^a=bf,c^a=cgjk,c^b=cfgij,d^a=djk,d^b=di,d^c=dgjk,e^a=ek,e^b=ej,f^a=fi,f^b=fi,f^c=fi,f^d=fj,f^e=fk,g^c=gk,h^b=hk,i^a=ij,i^b=ij,i^c=ij,i^d=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227158
maximal normal subgroup of	#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.7);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.11);;
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.7*F.10*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10*F.11);;
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);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, 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);;
gap> SetConjugate(coll, 9, 4, F.9*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);