#72227003

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225229
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	C₂×#33858	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #493217 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2137 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=d,d^2=g,e^2=hj,f^2=i,h^2=jk,j^2=k,b^a=be,c^b=cefi,d^a=djk,d^c=di,e^a=eh,e^b=ehjk,e^c=eh,e^d=ek,f^a=fjk,f^b=fjk,f^c=fi,g^a=gk,h^a=hj,h^b=hjk,h^c=hj,i^a=ik,i^b=ik,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	#72227037, #72227138, #72227167
maximal normal subgroup of	#72227037, #72227138, #72227167

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