#72226913

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#123275
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
socle	C₂	#72225188
prime radical	#72225339	C₂
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

837 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)(3,16)(4,22)(5,26)(6,15)(7,10)(8,20)(9,37)(11,30)(12,45)(13,24)(14,25)(17,49)(18,51)(19,38)(21,32)(23,42)(27,35)(28,40)(29,41)(31,50)(33,60)(34,52)(36,54)(39,64)(43,59)(44,61)(46,56)(47,57)(48,55)(53,63)(58,62);(1,3)(2,8)(4,17)(5,27)(6,18)(7,31)(9,38)(10,35)(11,44)(12,32)(13,46)(14,47)(15,36)(16,26)(19,24)(20,45)(21,25)(22,34)(23,57)(28,61)(29,56)(30,53)(33,43)(37,60)(39,48)(40,51)(41,50)(42,49)(52,62)(54,64)(55,59)(58,63);(2,9,10,39,13,40,43,58,15,42,11,12,22,26,14,41)(3,17,18,19)(5,28)(8,32,33,38,34,63,35,16,36,48,47,49,46,50,44,51)(20,56,54,52)(21,53,55,31)(23,37)(27,60,61,57)(29,62)(45,64)

Polycyclic presentation:

2^11|c^2=d,d^2=gik,e^2=g,f^2=h,g^2=i,h^2=jk,j^2=k,b^a=bf,c^a=cfjk,c^b=ce,d^a=dh,d^b=dg,e^a=ehj,e^b=egi,f^a=fhj,f^b=fhj,f^c=fk,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

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226924, #72226950, #72226961
maximal normal subgroup of	#72226924, #72226950, #72226961

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