#72319795

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	#72317727
commutator, Frattini, cosocle	C₂²×#127	C₂⁴
prime radical	C₂×#33557	C₂²
LCS	G ▷ C₂²×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#117 ◁ #34210 ◁ G
LECS	G ▷ C₂²×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

5551 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)(4,18,21,9)(5,10)(6,11)(7,26)(12,25)(13,32,39,45)(14,29)(15,43)(16,30)(17,31)(19,47,44,28)(20,46,49,33)(22,37,27,34)(23,51)(24,35)(36,53)(38,57,60,63)(40,55)(41,61)(42,56)(48,64,62,54)(50,59,52,58);(1,3)(2,8)(4,19,22,13)(5,14)(6,17)(7,16,25,15)(9,32,34,47)(10,29)(11,31)(12,30,26,43)(18,45,37,28)(20,48,50,38)(21,44,27,39)(23,42,53,41)(24,40)(33,57,58,64)(35,55)(36,56,51,61)(46,63,59,54)(49,62,52,60);(1,4,6,21)(2,9,11,18)(3,13)(5,20)(7,27,25,22)(8,28)(10,33)(12,37,26,34)(14,38,40,60)(15,19)(16,44)(17,39)(23,52)(24,49)(29,54,55,64)(30,32)(31,47)(35,46)(36,59)(41,48,42,62)(43,45)(50,53)(51,58)(56,57,61,63);(1,5,6,24)(2,10,11,35)(3,14,17,40)(4,20)(7,23,25,53)(8,29,31,55)(9,33)(12,36,26,51)(13,38)(15,41,16,42)(18,46)(19,48)(21,49)(22,50)(27,52)(28,54)(30,56,43,61)(32,57)(34,58)(37,59)(39,60)(44,62)(45,63)(47,64)

Polycyclic presentation:

2^11|a^2=f,b^2=h,c^2=e,d^2=fk,i^2=k,b^a=bfh,c^a=cg,c^b=cfgk,d^c=dej,e^b=ek,e^d=ek,f^c=fk,g^a=gk,g^b=gi,h^c=hik,h^g=hjk,i^a=ij,i^b=ij,i^c=ik,i^g=ik,i^h=ik,j^b=jk,j^c=jk

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

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