#72225455

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225454
commutator	C₄×C₃₂	C₂²×C₄
Frattini, cosocle	C₄×M₆₄	C₂³
prime radical	#33981	Q₈
LCS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ C₈×M₆₄ ◁ G
LECS	G ▷ C₄×M₆₄ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

467 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=e,c^2=fh,d^2=gi,e^2=h,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bd,c^a=cik,c^b=cf,d^a=dg,d^b=dgj,e^a=ej,e^d=ek,f^a=fj,f^b=fh,g^a=gi,g^b=gik,h^a=hk,i^a=ij,i^b=ij,j^a=jk,j^b=jk

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225464, #72225510, #72225515
maximal normal subgroup of	#72225464, #72225510, #72225515

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