#72343446

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72285149
commutator	C₂×M₃₂	C₂³×C₄
Frattini, cosocle	C₄×M₃₂	C₂⁴
prime radical	#151182	C₄
LCS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #479966 ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1628 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=k,d^2=g,e^2=h,f^2=ik,i^2=j,j^2=k,b^a=bhjk,d^a=df,d^b=dfi,d^c=dg,e^d=ej,f^a=fi,f^b=fij,f^c=fh,f^d=fhijk,f^e=fk,g^a=ghk,g^b=ghk,g^f=gk,h^d=hk,i^a=ij,i^b=ijk,i^d=ijk,j^a=jk,j^b=jk,j^d=jk

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343465
maximal normal subgroup of	#72343465

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