#72319743

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72317683
commutator, Frattini, cosocle	C₂×#535	C₂⁴
prime radical	#72318458	C₂
LCS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35327 ◁ G
LECS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

3981 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:

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

Polycyclic presentation:

2^11|b^2=e,c^2=ghj,d^2=hjk,i^2=k,b^a=bf,c^a=cg,c^b=cfh,d^a=dhjk,d^b=dehijk,d^c=dk,e^a=eh,e^c=eh,f^b=fh,f^c=fhik,f^d=fhj,g^b=ghi,g^e=gk,g^f=gjk,h^c=hk,i^a=ij,i^c=ijk,i^f=ik,i^g=ik,j^b=jk,j^c=jk

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320243
maximal normal subgroup of	#72320243

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