#72341236

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72237742
commutator, Frattini, cosocle	#1781	C₂⁴
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #21749 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

1977 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=ehijk,c^2=gk,d^2=hi,e^2=k,f^2=hjk,b^a=be,c^a=cgj,c^b=cf,d^a=dijk,d^b=defik,d^c=di,e^a=ek,e^b=ejk,e^c=eg,e^d=egij,f^a=fgj,f^b=fgh,f^c=fhij,f^d=fgh,f^e=fijk,g^b=gi,g^d=gj,g^f=gjk,h^a=hjk,h^b=hj,h^c=hk,h^d=hjk,h^e=hk,i^c=ij,i^d=ijk,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, 1, F.11);;
gap> SetPower(coll, 2, F.5*F.8*F.9*F.10*F.11);;
gap> SetPower(coll, 3, F.7*F.11);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 6, F.8*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.9*F.11);;
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.10*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 5, 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, 6, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);