#72227074

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225256
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	C₂×#2246	C₂×C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #129051 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

769 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=f,d^2=g,e^2=h,f^2=ik,h^2=j,i^2=k,j^2=k,b^a=be,c^a=cijk,c^b=cefi,d^b=djk,d^c=dk,e^a=eh,e^b=ehj,e^c=ehjk,e^d=ek,f^a=fk,f^c=fi,g^b=gk,h^a=hj,h^b=hjk,h^c=hjk,i^c=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227077, #72227167, #72227175
maximal normal subgroup of	#72227077, #72227167, #72227175

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.4);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.10*F.11);;
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*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 3, 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);