#72226915

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	#72225339	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

715 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)(3,22)(4,12)(5,32)(6,16)(7,39)(8,41)(9,17)(10,19)(11,21)(13,29)(14,37)(15,59)(18,33)(20,42)(23,46)(24,54)(25,44)(26,52)(27,61)(28,53)(30,48)(31,50)(34,51)(35,57)(36,58)(38,56)(40,60)(43,45)(47,63)(49,62)(55,64);(1,3)(2,12)(4,6)(5,29)(7,23)(8,40)(9,28)(10,43)(11,27)(13,51)(14,45)(15,54)(16,53)(17,47)(18,46)(19,49)(20,48)(21,50)(22,39)(24,34)(25,33)(26,42)(30,59)(31,35)(32,61)(36,63)(37,64)(38,62)(41,55)(44,56)(52,58)(57,60);(1,4,5,28,33,63,59,26,10,30,36,24,8,29,6,27,11,31,7,40,34,64,38,43,42,62,37,25,9,23,35,3)(2,13,14,52,51,12,16,44,19,55,32,61,17,48,56,53,21,46,58,45,18,50,57,54,20,47,39,22,41,49,15,60)

Polycyclic presentation:

2^11|c^2=d,d^2=gik,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, #72226963
maximal normal subgroup of	#72226924, #72226936, #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, 3, F.4);;
gap> SetPower(coll, 4, F.7*F.9*F.11);;
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);