#72227055

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227054
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	#33713	C₂×C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #129051 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=f,b^2=d,c^2=dfhijk,d^2=g,e^2=hjk,f^2=ijk,h^2=jk,j^2=k,b^a=be,c^a=cfik,c^b=cefij,d^a=dj,d^c=di,e^a=eh,e^b=ehjk,e^c=ehj,e^d=ek,f^b=fk,f^c=fijk,g^a=gk,h^a=hj,h^b=hjk,h^c=hjk,i^a=ik,i^b=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227066, #72227168, #72227179
maximal normal subgroup of	#72227066, #72227168, #72227179

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