#72341206

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72237594
commutator	#379	C₂³×C₄
Frattini, cosocle	#1781	C₂⁴
prime radical	#72341066	C₂
LCS	G ▷ #379 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ #6374210 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

3775 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)(3,14)(4,11)(5,23)(6,27)(7,13)(8,19)(9,22)(10,26)(12,31)(15,44)(16,33)(17,34)(18,42)(20,37)(21,38)(24,39)(25,40)(28,35)(29,41)(30,43)(32,47)(36,53)(45,51)(46,57)(48,58)(49,61)(50,60)(52,59)(54,63)(55,62)(56,64);(1,3,4,17)(2,8,11,34)(5,16,20,49)(6,28,29,15)(7,19,22,53)(9,36,13,14)(10,33,39,59)(12,42,38,44)(18,21,47,31)(23,45,37,61)(24,52,26,51)(25,56,54,46)(27,35,41,32)(30,50,55,48)(40,64,63,58)(43,60,62,57);(1,4)(2,9)(3,15)(5,24)(6,29,31,21)(7,22)(8,32)(10,39)(11,13)(12,41,27,38)(14,44)(16,48,51,46)(17,18)(19,47)(20,26)(23,37)(25,54,30,55)(28,53)(33,58,45,57)(34,42)(35,36)(40,62,43,63)(49,50,52,56)(59,64,61,60);(1,5)(2,10)(3,16,19,51)(4,20,22,24)(6,30,31,25)(7,26)(8,33,14,45)(9,37,11,39)(12,43,27,40)(13,23)(15,46,47,48)(17,52)(18,50)(21,55)(28,56)(29,54)(32,57,44,58)(34,61)(35,60)(36,59)(38,63)(41,62)(42,64)(49,53)

Polycyclic presentation:

2^11|b^2=e,c^2=gj,d^2=hijk,f^2=h,b^a=bhijk,c^a=cjk,c^b=cf,d^a=dk,d^b=dghij,d^c=dgijk,e^a=ej,e^c=eg,e^d=eij,f^a=fjk,f^b=fgh,f^c=fhi,f^d=fij,f^e=fijk,g^b=gi,g^d=gj,g^f=gjk,h^b=hjk,h^c=hk,h^d=hk,h^e=hk,i^c=ij,i^d=ik,i^f=ik,j^b=jk

Conjugacy classes & character table

65 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.5);;
gap> SetPower(coll, 3, F.7*F.10);;
gap> SetPower(coll, 4, F.8*F.9*F.10*F.11);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9*F.10*F.11);;
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.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
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, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*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.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);