#72226996

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#13617
commutator	C₂×C₃₂	C₂²×C₁₆
Frattini, cosocle	C₁₆×C₃₂	C₂³
socle	C₂	#72226995
prime radical	#33981	C₁₆
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ C₄×M₆₄ ◁ C₈×M₆₄ ◁ C₁₆×M₆₄ ◁ G
LECS	G ▷ C₁₆×C₃₂ ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

678 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)(3,8)(4,10)(5,12)(6,14)(7,9)(11,41)(13,45)(15,47)(16,26)(17,31)(18,27)(19,28)(20,29)(21,38)(22,39)(23,33)(24,37)(25,35)(30,61)(32,50)(34,62)(36,54)(40,56)(42,51)(43,63)(44,64)(46,55)(48,58)(49,60)(52,57)(53,59);(2,8,10,27,12,28,38,42,14,29,39,57,37,59,36,34,9,31,33,58,35,60,46,44,16,32,40,43,15,30,13,11);(2,9)(3,17)(10,33)(12,35)(13,36)(14,16)(15,37)(18,48)(19,49)(20,50)(38,46)(39,40)(41,62)(51,64)(52,63)(53,61)

Polycyclic presentation:

2^12|b^2=d,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,k^2=l,b^a=be,c^a=cl,c^b=cj,d^a=dg,e^a=egikl,e^c=el,f^a=fi,g^a=gikl,h^a=hk,i^a=ikl,j^a=jl,k^a=kl

Conjugacy classes & character table

520 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227029, #72227093, #72227136
maximal normal subgroup of	#72227029, #72227093, #72227136

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.12);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9*F.11*F.12);;
gap> SetConjugate(coll, 5, 3, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.11*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);