#72342966

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	#72320606
commutator, Frattini, cosocle	C₂×#504	C₂⁴
prime radical	#72320482	C₂
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

4953 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)(5,23)(7,28)(9,31)(11,37)(12,29)(14,45)(15,33)(16,38)(17,42)(20,43)(21,51)(22,36)(24,40)(25,44)(26,30)(27,53)(32,60)(34,59)(35,61)(39,64)(41,55)(46,62)(47,56)(48,49)(50,63)(52,58)(54,57);(1,2,6,8)(4,19,18,10)(5,24,25,11)(7,12,29,28)(13,42)(21,35,27,39)(22,49,48,36)(23,37,44,40)(26,56)(31,33)(38,55)(43,45)(51,64,53,61)(52,57)(59,60)(62,63);(3,14)(5,25)(9,32)(11,24)(13,43)(15,34)(16,46)(17,20)(21,27)(23,44)(26,57)(30,58)(31,59)(33,60)(35,39)(37,40)(38,63)(41,50)(42,45)(47,54)(51,53)(52,56)(55,62)(61,64);(1,3,6,17)(2,9,8,15)(4,20)(5,26,25,56)(7,30,29,47)(10,34)(11,38,24,55)(12,41,28,16)(13,44,42,23)(14,18)(19,32)(21,52)(22,54)(27,57)(31,40,33,37)(35,62)(36,46)(39,63)(43,51)(45,53)(48,58)(49,50)(59,61)(60,64)

Polycyclic presentation:

2^11|b^2=h,d^2=fjk,f^2=k,g^2=ij,i^2=jk,j^2=k,b^a=bk,c^a=chk,d^a=dg,d^b=df,d^c=de,e^a=eh,e^b=eh,e^d=ehk,f^a=fk,f^c=fhk,f^d=fk,f^e=fk,g^a=gi,g^b=gk,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, 2, F.8);;
gap> SetPower(coll, 4, F.6*F.10*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.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, 1, F.6*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.11);;
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);