#72318472

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72317748
commutator	C₂×#127	C₂³×C₄
Frattini, cosocle	C₂×#535	C₂⁴
prime radical	#72317462	C₂
LCS	G ▷ C₂×#127 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ C₂×#3498 ◁ G
LECS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

5306 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)(3,9)(4,10)(5,20)(6,27)(7,12)(8,31)(11,24)(13,30)(14,32)(15,33)(16,43)(17,49)(18,37)(19,38)(21,39)(22,56)(23,55)(25,42)(26,41)(28,36)(29,34)(35,51)(40,52)(44,63)(45,60)(46,61)(47,64)(48,54)(50,59)(53,58)(57,62);(1,3)(2,9)(4,14)(5,21)(6,7)(8,29)(10,32)(11,39)(12,27)(13,43)(15,26)(16,30)(17,50)(18,45)(19,28)(20,54)(22,44)(23,25)(24,48)(31,34)(33,42)(35,63)(36,38)(37,60)(40,59)(41,55)(46,53)(47,52)(49,64)(51,57)(56,62)(58,61);(1,4)(2,10)(3,14)(5,22,25,17)(6,28)(7,19)(8,18,30,53)(9,32)(11,40,41,35)(12,38)(13,37,31,58)(15,47,48,57)(16,46,29,45)(20,49,42,56)(21,44,23,50)(24,51,26,52)(27,36)(33,62,54,64)(34,61,43,60)(39,59,55,63);(1,5,6,23,7,24,3,15)(2,11,9,33,12,20,27,55)(4,17,14,44,19,52,28,57)(8,26,29,48,30,25,16,21)(10,35,36,64,38,56,32,59)(13,42,34,39,31,41,43,54)(18,51,46,47,53,22,45,50)(37,49,60,63,58,40,61,62)

Polycyclic presentation:

2^11|c^2=g,d^2=e,e^2=k,i^2=k,b^a=bhk,c^a=cg,d^a=df,d^b=dhij,d^c=dfk,e^a=eh,e^b=ek,e^c=eh,f^b=fj,f^c=fhik,f^d=fh,f^e=fk,g^d=ghi,g^f=gjk,h^d=hk,i^a=ij,i^b=ik,i^c=ij,i^f=ik,i^g=ik,j^c=jk,j^d=jk

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319061, #72320243
maximal normal subgroup of	#72319061, #72320243

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