#72318854

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	#72240171
commutator, Frattini, cosocle	C₂×#535	C₂⁴
LCS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35527 ◁ G
LECS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

5018 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)(3,8)(4,18)(5,10)(6,12)(7,11)(9,21)(13,29)(14,45)(15,31)(16,32)(17,41)(19,37)(20,42)(22,53)(23,52)(24,57)(25,58)(26,39)(27,34)(28,40)(30,47)(33,62)(35,56)(36,55)(38,50)(43,64)(44,59)(46,61)(48,63)(49,51)(54,60);(3,13)(4,19)(6,28)(8,29)(9,18)(12,40)(14,46)(20,27)(21,22)(23,55)(24,56)(25,50)(30,51)(33,63)(34,42)(35,52)(36,57)(37,53)(38,58)(43,54)(45,60)(47,48)(49,62)(61,64);(1,3,7,13)(2,8,11,29)(4,14,24,48)(5,25)(6,16)(9,33,36,60)(10,38)(12,41)(15,20,44,27)(17,28)(18,51,57,64)(19,54,56,62)(21,46,55,47)(22,43,23,49)(26,50)(30,35,61,37)(31,42,59,34)(32,40)(39,58)(45,53,63,52);(1,4,20,55,7,22,27,56)(2,9,34,52,11,37,42,57)(3,14,15,47,13,43,44,62)(5,21,28,23,26,19,6,24)(8,30,59,64,29,33,31,45)(10,18,12,36,39,53,40,35)(16,49,50,54,17,48,25,46)(32,61,58,51,41,60,38,63)

Polycyclic presentation:

2^11|c^2=gk,d^2=efhijk,i^2=k,b^a=bhj,c^a=cg,c^b=ck,d^a=de,d^b=def,d^c=dfijk,e^b=eh,e^c=ei,e^d=ehjk,f^a=fhk,f^b=fh,f^c=fhik,f^d=fhj,g^d=ghijk,g^e=gjk,g^f=gj,h^c=hk,i^a=ij,i^b=ik,i^c=ij,i^e=ik,i^f=ik,i^g=ik,j^c=jk,j^d=jk

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319061
maximal normal subgroup of	#72319061

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