#72258161

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#72225319
commutator	C₄×C₁₆	C₂×C₄×C₈
Frattini, cosocle	C₈×#346	C₂³
socle	C₂	#72258160
prime radical	C₂×#508	#97
LCS	G ▷ C₄×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₁₆ ◁ C₄×#1398 ◁ C₈×#1398 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1114 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:

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

Polycyclic presentation:

2^12|a^2=l,b^2=d,c^2=dehj,d^2=g,e^2=h,f^2=i,h^2=j,i^2=k,k^2=l,b^a=bf,c^a=cfil,c^b=cfhl,d^a=dkl,d^c=dj,e^a=ei,e^b=eij,e^c=ej,f^a=fikl,f^b=fil,f^c=fkl,f^d=fl,g^a=gl,h^a=hk,h^b=hk,i^a=ikl,i^b=ik,i^c=il,j^a=jl,j^b=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258194
maximal normal subgroup of	#72258194

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