#72341200

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72237728
commutator, Frattini, cosocle	#1781	C₂⁴
prime radical	#72285594	C₂
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂⁶ ◁ #21746 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

2086 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)(3,8)(4,9)(5,10)(6,12)(7,13)(11,21)(14,46)(15,49)(16,35)(17,43)(18,34)(19,36)(20,37)(22,38)(23,39)(24,40)(25,41)(26,33)(27,42)(28,44)(29,45)(30,48)(31,52)(32,53)(47,58)(50,59)(51,60)(54,62)(55,64)(56,61)(57,63);(1,3,7,18)(2,8,13,34)(4,16,21,53)(5,22,25,56)(6,26,29,17)(9,35,11,32)(10,38,41,61)(12,43,45,33)(14,27,48,20)(15,23,52,19)(24,50,28,47)(30,42,46,37)(31,36,49,39)(40,58,44,59)(51,57,55,54)(60,62,64,63);(1,4)(2,9)(3,14)(5,23,25,19)(6,27)(7,21)(8,30)(10,39,41,36)(11,13)(12,42)(15,50)(16,26,53,17)(18,48)(20,29)(22,55,56,51)(24,54,28,57)(31,59)(32,33,35,43)(34,46)(37,45)(38,60,61,64)(40,62,44,63)(47,52)(49,58);(1,5)(2,10)(3,15)(4,19)(6,28,29,24)(7,25)(8,31)(9,36)(11,39)(12,44,45,40)(13,41)(14,47,48,50)(16,22)(17,51,26,55)(18,52)(20,54,27,57)(21,23)(30,58,46,59)(32,38)(33,60,43,64)(34,49)(35,61)(37,62,42,63)(53,56)

Polycyclic presentation:

2^11|b^2=k,c^2=gh,d^2=h,e^2=h,f^2=hik,b^a=bh,c^a=cjk,c^b=cf,d^a=djk,d^b=de,d^c=di,e^a=ek,e^b=eh,e^c=eg,e^d=eh,f^a=fjk,f^b=fhik,f^c=fh,f^d=fgi,f^e=fi,g^b=gi,g^c=gjk,g^d=gj,g^e=gjk,g^f=gj,h^c=hjk,h^f=hk,i^c=ij,i^d=ijk,i^e=ik,i^f=ik,j^b=jk

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

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