#72225593

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×D₁₆²
commutator, Frattini, cosocle	C₈×C₁₆	C₂⁵
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^12|a^2=l,c^2=fh,d^2=gi,f^2=hj,g^2=ik,h^2=j,i^2=kl,k^2=l,c^a=cg,c^b=cf,d^a=dg,d^b=dg,e^a=el,e^d=el,f^a=fi,f^b=fh,g^a=gi,g^b=gi,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

274 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226982, #72227093, #72227115
maximal normal subgroup of	#72226982, #72227093, #72227115

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