#72340229

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	#72216457
commutator, Frattini, cosocle	#1781	C₂⁴
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #20586 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

3688 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)(9,32)(10,30)(11,31)(12,25)(14,43)(15,50)(16,45)(17,46)(18,44)(19,22)(29,35)(33,56)(34,58)(36,57)(38,41)(40,42)(47,61)(48,49)(51,64)(52,62)(53,63)(54,55);(1,2,7,8)(3,14,17,49)(4,10,21,30)(5,11,24,31)(6,25,28,12)(9,33,29,16)(13,43,46,48)(15,51,18,47)(19,37,22,39)(20,38,26,41)(23,40,27,42)(32,56,35,45)(34,53,36,52)(44,61,50,64)(54,60,55,59)(57,62,58,63);(1,3,6,13)(2,9,12,35)(4,18,20,44)(5,16,27,56)(7,17,28,46)(8,29,25,32)(10,36,38,58)(11,14,42,43)(15,26,50,21)(19,52,55,62)(22,53,54,63)(23,45,24,33)(30,34,41,57)(31,49,40,48)(37,51,60,61)(39,47,59,64);(1,4,7,21)(2,10,8,30)(3,15,17,18)(5,22,24,19)(6,26,28,20)(9,34,29,36)(11,39,31,37)(12,41,25,38)(13,44,46,50)(14,47,49,51)(16,53,33,52)(23,54,27,55)(32,57,35,58)(40,59,42,60)(43,61,48,64)(45,63,56,62)

Polycyclic presentation:

2^11|b^2=k,c^2=fik,d^2=k,e^2=h,b^a=bjk,c^a=cf,c^b=ce,d^a=dijk,d^b=dhj,d^c=dijk,e^a=eg,e^b=eh,e^c=eghijk,e^d=ek,f^b=fgij,f^d=fk,f^e=fijk,g^b=gj,g^c=gi,g^d=gk,g^e=gj,h^a=hj,h^c=hjk,h^f=hk,i^b=ij,i^e=ik,j^c=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340290, #72340332, #72340478, …
maximal normal subgroup of	#72340290, #72340332, #72340478, …

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