#72284775

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
prime radical	#72237721	C₂
LCS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #20585 ◁ G
LECS	G ▷ #1781 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

3559 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:

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

Polycyclic presentation:

2^11|b^2=ghk,c^2=i,d^2=hik,e^2=h,f^2=k,b^a=bghjk,c^a=ce,c^b=cf,d^a=dg,d^b=dgj,d^c=dhij,e^a=eh,e^b=eg,e^c=ehj,e^d=eik,f^a=fg,f^b=fik,f^d=fij,f^e=fij,g^b=gjk,g^c=gi,g^d=gj,g^e=gjk,h^b=hjk,h^d=hk,h^f=hk,i^a=ij,i^b=ik,i^d=ik,i^e=ik,j^c=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286327
maximal normal subgroup of	#72286327

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