#72319891

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72240175
commutator, Frattini, cosocle	C₂×#535	C₂⁴
prime radical	#72318522	C₂
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

4615 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)(4,17)(5,10)(6,12)(7,11)(9,21)(13,30)(14,46)(15,32)(16,33)(18,38)(19,56)(20,41)(22,53)(23,51)(24,58)(25,40)(26,42)(27,34)(28,52)(29,36)(31,48)(35,47)(37,55)(39,50)(43,54)(44,63)(45,61)(49,64)(57,60)(59,62);(3,13)(4,18)(6,26)(8,30)(9,17)(12,42)(14,47)(19,57)(20,29)(21,22)(23,54)(24,50)(28,55)(31,56)(35,63)(36,41)(37,51)(38,53)(39,58)(43,52)(44,59)(46,62)(48,49)(60,64);(1,3)(2,8)(4,19,28,44)(5,24,25,50)(6,27,26,16)(7,13)(9,35,43,31)(10,39,40,58)(11,30)(12,33,42,34)(14,22,49,23)(15,20)(17,46,52,64)(18,48,55,47)(21,57,54,59)(29,45)(32,41)(36,61)(37,60,38,62)(51,56,53,63);(1,4,20,55,7,22,29,54)(2,9,36,52,11,38,41,51)(3,14,15,48,13,44,45,57)(5,21,6,28,25,18,26,23)(8,31,61,63,30,60,32,46)(10,17,42,37,40,53,12,43)(16,19,24,59,27,49,50,47)(33,35,39,56,34,62,58,64)

Polycyclic presentation:

2^11|c^2=ghj,d^2=efi,i^2=k,b^a=bhj,c^a=cg,c^b=ck,d^a=de,d^b=def,d^c=dfh,e^b=eh,e^c=ei,e^d=ehjk,f^a=fhk,f^b=fh,f^c=fhi,f^d=fhj,g^d=ghik,g^e=gjk,g^f=gj,h^c=hk,i^a=ij,i^b=ik,i^c=ij,i^d=ik,i^e=ik,i^f=ik,i^g=ik,j^c=jk,j^d=jk

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320243
maximal normal subgroup of	#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*F.8*F.10);;
gap> SetPower(coll, 4, F.5*F.6*F.9);;
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.8);;
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);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.9*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, 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, 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);