#72227092

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#123275
commutator, Frattini, cosocle	C₁₆²	C₂⁴
LCS	G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×C₁₆² ◁ G
LECS	G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

3306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^12|e^2=gi,f^2=hj,g^2=ik,h^2=jl,i^2=k,j^2=l,c^a=cf,c^b=ce,d^c=dk,e^a=eh,e^b=eg,e^c=eg,f^a=fh,f^b=fh,f^c=fh,g^a=gj,g^b=gi,g^c=gi,h^a=hj,h^b=hj,h^c=hj,i^a=il,i^b=ik,i^c=ik,j^a=jl,j^b=jl,j^c=jl

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227169
maximal normal subgroup of	#72227169
maximal quotient of	#72227093

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