#72319755

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	#72317955
commutator, Frattini, cosocle	C₂²×#127	C₂⁴
prime radical	#6442565	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

4857 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)(4,9)(5,21,23,10)(6,25)(7,11)(12,27)(13,28)(14,35,42,52)(15,30)(16,44)(17,31)(18,32,47,53)(19,33)(20,50)(22,54,43,29)(24,37,26,36)(34,49)(38,60,41,64)(39,57)(40,58)(45,56)(46,59,63,55)(48,61,51,62);(1,3)(2,8)(4,13)(5,22,24,14)(6,17,27,16)(7,15)(9,28)(10,35,36,54)(11,30)(12,31,25,44)(18,46,48,41)(19,40)(20,39,49,45)(21,52,37,29)(23,43,26,42)(32,60,61,55)(33,58)(34,57,50,56)(38,47,63,51)(53,64,62,59);(1,4,7,19)(2,9,11,33)(3,13,15,40)(5,18)(6,20,27,49)(8,28,30,58)(10,32)(12,34,25,50)(14,41)(16,45,17,39)(21,53)(22,46)(23,47)(24,48)(26,51)(29,59)(31,57,44,56)(35,60)(36,61)(37,62)(38,42)(43,63)(52,64)(54,55);(1,5,7,23)(2,10,11,21)(3,14,15,42)(4,18)(6,26,27,24)(8,29,30,54)(9,32)(12,37,25,36)(13,38)(16,22,17,43)(19,47)(20,51)(28,55)(31,35,44,52)(33,53)(34,62)(39,46)(40,41)(45,63)(48,49)(50,61)(56,60)(57,64)(58,59)

Polycyclic presentation:

2^11|a^2=f,b^2=h,c^2=fk,d^2=ej,i^2=k,b^a=bfh,c^a=cf,d^a=dfg,d^b=dfg,d^c=de,e^b=ek,e^c=ek,e^d=ek,f^d=fk,g^a=gk,g^b=gi,g^c=gk,g^d=gk,h^d=hik,h^g=hjk,i^a=ij,i^b=ij,i^d=ik,i^g=ik,i^h=ik,j^b=jk,j^d=jk

Conjugacy classes & character table

83 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.6*F.11);;
gap> SetPower(coll, 4, F.5*F.10);;
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.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 4, 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, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 4, 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, 4, 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, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);