#72226916

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, socle	C₂	#72226914
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	C₂×#151634	C₂
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

751 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)(4,26)(5,31)(6,10)(7,39)(8,17)(9,19)(11,43)(12,32)(13,27)(14,36)(15,34)(16,28)(18,35)(21,55)(22,58)(23,51)(24,54)(25,48)(29,52)(30,42)(33,44)(37,41)(38,64)(40,57)(45,62)(46,61)(47,56)(49,59)(50,60)(53,63);(2,10)(3,21)(4,27)(5,32)(6,36)(7,28)(8,35)(12,18)(13,52)(14,16)(15,31)(17,42)(19,26)(20,47)(22,59)(23,62)(24,63)(25,33)(29,34)(30,37)(38,60)(39,41)(40,45)(43,61)(44,64)(46,53)(48,54)(49,50)(55,58)(56,57);(1,3,4,22,28,60,5,33,8,24,30,61,34,43,36,63,9,25,6,38,29,59,37,21,35,56,32,62,7,23,27,57)(2,11,12,44,52,20,14,45,17,49,26,53,39,54,41,58,19,51,42,55,16,48,13,46,18,50,10,40,15,47,31,64)

Polycyclic presentation:

2^11|c^2=d,d^2=gi,e^2=gk,f^2=hj,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=bf,c^a=chk,c^b=ce,d^a=djk,d^b=dg,e^a=ejk,e^b=egik,e^c=ek,f^a=fh,f^b=fh,f^c=fk,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	#72226924, #72226936, #72226958
maximal normal subgroup of	#72226924, #72226936, #72226958

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