#72227146

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	#72227144
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	#512851	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

1488 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:

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

Polycyclic presentation:

2^11|a^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=cfijk,d^a=djk,d^b=dik,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, 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.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.11);;
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);