#72321336

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72320532
commutator, Frattini, cosocle	C₂×#504	C₂⁴
prime radical	#72320653	C₂
LCS	G ▷ C₂×#504 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ #7622775 ◁ G
LECS	G ▷ C₂×#504 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5434 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:

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

Polycyclic presentation:

2^11|a^2=h,c^2=efgik,d^2=fj,e^2=k,f^2=ijk,g^2=j,i^2=jk,j^2=k,b^a=bh,c^a=ce,c^b=cefik,d^a=dehk,d^b=defik,d^c=deghij,e^c=eh,e^d=ek,f^a=fk,f^b=fi,g^a=gh,g^b=ghjk,g^c=gh,g^d=gk,g^e=gk,h^c=hk,h^d=hk,i^b=ij,j^b=jk

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343072
maximal normal subgroup of	#72343072

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