#72342917

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72320641
commutator, Frattini, cosocle	C₂×#504	C₂⁴
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

5164 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:

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

Polycyclic presentation:

2^11|c^2=ek,d^2=fh,e^2=k,f^2=k,g^2=ij,i^2=jk,j^2=k,b^a=bhk,c^a=cg,c^b=ce,d^a=dghk,d^b=de,d^c=defk,e^a=eh,e^b=ek,e^d=eh,f^a=fk,f^b=fhk,f^c=fhk,f^d=fk,f^e=fk,g^a=gi,g^b=gh,g^c=ghi,g^d=gi,g^e=gk,h^c=hk,h^d=hk,i^a=ij,i^c=ij,i^d=ij,j^a=jk,j^c=jk,j^d=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, 3, F.5*F.11);;
gap> SetPower(coll, 4, F.6*F.8);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.9*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*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.6*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
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, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);