#72226834

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72226833
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	C₂×#508	D₈
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

555 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)(3,20)(4,28)(5,33)(6,36)(7,41)(8,18)(9,44)(10,14)(11,19)(12,23)(13,42)(15,39)(16,29)(17,43)(21,60)(22,47)(24,56)(25,52)(26,59)(27,46)(30,49)(31,54)(32,45)(34,51)(35,50)(37,64)(38,57)(40,55)(48,63)(53,62)(58,61);(1,3,5,21,11,27,10,26)(2,12,14,45,19,49,33,28)(4,29,30,40,32,39,23,6)(7,42,34,63,38,31,37,22)(8,25,35,61,43,62,9,24)(13,50,47,18,54,44,48,17)(15,56,55,53,16,58,36,52)(20,51,59,41,46,64,60,57);(1,4,7,25,8,31,40,3,5,23,34,24,35,63,39,26,11,32,38,62,43,42,6,27,10,30,37,61,9,22,29,21)(2,13,51,60,17,12,16,53,33,47,57,20,44,45,55,58,19,54,64,59,18,49,15,52,14,48,41,46,50,28,36,56)

Polycyclic presentation:

2^11|b^2=d,c^2=egik,d^2=k,e^2=gik,f^2=hj,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=bf,c^a=chjk,c^b=ce,d^a=dk,d^c=dk,e^a=ejk,e^b=eg,f^a=fh,f^b=fhk,g^a=gk,g^b=gi,h^a=hj,h^b=hj,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

158 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226838, #72226924, #72226936
maximal normal subgroup of	#72226838, #72226924, #72226936

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