#72342936

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	#72320508
commutator, Frattini, cosocle	C₂×#504	C₂⁴
LCS	G ▷ C₂×#504 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ #7622779 ◁ G
LECS	G ▷ C₂×#504 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5775 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:

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

Polycyclic presentation:

2^11|d^2=f,g^2=ij,i^2=jk,j^2=k,c^a=chk,d^a=dg,d^b=df,d^c=de,e^a=eh,e^b=eh,e^d=ehk,f^c=fhk,f^e=fk,g^a=gi,g^c=gh,g^d=gi,g^e=gk,h^d=hk,i^a=ij,i^d=ij,j^a=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, 4, F.6);;
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, 3, 1, F.3*F.8*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, 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, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);