#72225515

order	4096
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225514
commutator	C₈×C₃₂	C₂²×C₄
Frattini, cosocle	C₈×M₆₄	C₂³
prime radical	#72225430	C₂
LCS	G ▷ C₈×C₃₂ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ #72216226 ◁ G
LECS	G ▷ C₈×M₆₄ ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

832 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:

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

Polycyclic presentation:

2^12|a^2=l,c^2=e,d^2=gi,e^2=h,f^2=h,g^2=ik,h^2=j,i^2=kl,k^2=l,b^a=bd,c^a=cgk,c^b=cf,d^a=dg,d^b=dg,d^c=dkl,e^a=eik,e^b=eh,e^d=el,f^a=fikl,f^b=fhj,g^a=gi,g^b=gi,g^c=gl,h^a=hk,h^b=hj,i^a=ik,i^b=ik,j^a=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225550
maximal normal subgroup of	#72225550

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