#72225587

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	C₄	#123275
commutator, Frattini, cosocle	C₈×C₁₆	C₂⁴
socle	C₂	C₂×#123275
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2008 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:

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

Polycyclic presentation:

2^11|a^2=k,c^2=eg,e^2=gi,f^2=hj,g^2=i,h^2=jk,j^2=k,b^a=bf,c^a=cf,c^b=ce,d^a=dk,d^b=dk,e^a=eh,e^b=eg,f^a=fh,f^b=fh,g^a=gj,g^b=gi,h^a=hj,h^b=hj,i^a=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225593, #72226961, #72226963
maximal normal subgroup of	#72225593, #72226961, #72226963

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