#72284479

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	#72234744
commutator, Frattini, cosocle	#2309	C₂⁴
prime radical	#72283717	C₂
LCS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35626 ◁ G
LECS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

4284 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)(3,13)(4,9,20,35)(5,10,24,38)(6,25)(7,12)(8,16)(11,28)(14,45,48,42)(15,49)(17,44)(18,33)(19,34,26,40)(21,31)(22,36)(23,37,27,41)(29,47,39,52)(30,50)(32,56)(43,51)(46,61,54,62)(53,60)(55,59)(57,64,58,63);(3,14)(8,29)(11,25)(13,42)(15,46)(16,52)(17,48)(19,26)(21,47)(30,57)(31,39)(32,58)(37,41)(43,62)(44,45)(49,61)(50,64)(51,54)(53,55)(56,63);(1,3)(2,8,12,31)(4,15,20,51)(5,21)(6,14)(7,17)(9,32)(10,13,38,44)(11,39,25,29)(16,24)(18,50,22,56)(19,54,26,46)(23,47)(27,52)(28,48)(30,35)(33,43)(34,58)(36,49)(37,45,41,42)(40,57)(53,63,55,64)(59,62)(60,61);(1,4,7,20)(2,9)(3,15)(5,22)(6,26)(8,30,31,32)(10,36,38,33)(11,40,25,34)(12,35)(13,43)(14,46,48,54)(16,50,21,56)(17,51)(18,24)(19,28)(23,55,27,53)(29,57)(37,60)(39,58)(41,59)(42,61,45,62)(44,49)(47,64)(52,63)

Polycyclic presentation:

2^11|a^2=g,c^2=hik,d^2=hj,e^2=hik,f^2=hik,b^a=bgik,c^a=ce,c^b=cf,d^a=dghjk,d^b=dgij,d^c=dhij,e^a=eh,e^b=ejk,e^c=ehi,e^d=eijk,f^a=fij,f^b=fhik,f^c=fhik,f^d=fijk,f^e=fk,g^b=gjk,g^c=gik,g^d=gj,g^e=gj,g^f=gjk,h^a=hj,h^b=hjk,h^e=hk,h^f=hk,i^a=ijk,i^b=ijk,i^d=ik,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285855, #72286183
maximal normal subgroup of	#72285855, #72286183

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