#72323152

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#12067
commutator	C₂×C₁₆	C₂⁴×C₈
Frattini, cosocle	#1311	C₂⁵
prime radical	C₂×#33982	C₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#131 ◁ #72231879 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 5 generators on 80 points:

(1,2,4,9,5,10,11,15,6,8,14,3,7,12,16,13)(17,18,21,27,22,29,34,30,23,31,46,48,47,32,45,63)(19,25,36,28,37,24,33,50,39,51,38,62,71,52,65,54)(20,40,43,73,44,61,66,75,42,26,55,57,78,59,67,60)(35,56,68,74,69,72,64,80,41,53,70,58,77,49,76,79);(18,24)(19,33)(21,45)(22,47)(25,29)(26,49)(27,62)(28,48)(30,50)(31,52)(32,51)(34,46)(35,64)(36,37)(38,71)(39,65)(40,72)(41,76)(43,67)(44,78)(53,59)(54,63)(55,66)(56,61)(57,74)(58,73)(60,79)(68,69)(70,77)(75,80);(1,3,11,2,7,15,4,12,6,9,16,8,5,13,14,10)(17,19,34,33,47,71,21,36,23,39,45,65,22,37,46,38)(18,25,30,50,32,52,27,28,31,51,63,54,29,24,48,62)(20,41,66,76,78,69,43,70,42,35,67,64,44,77,55,68)(26,56,60,80,61,49,57,74,40,53,75,79,59,72,73,58);(2,8)(3,9)(10,12)(13,15)(17,20)(18,26)(19,35)(21,43)(22,44)(23,42)(24,49)(25,53)(27,57)(28,58)(29,59)(30,60)(31,40)(32,61)(33,64)(34,66)(36,68)(37,69)(38,70)(39,41)(45,67)(46,55)(47,78)(48,73)(50,79)(51,56)(52,72)(54,80)(62,74)(63,75)(65,76)(71,77);(20,42)(26,40)(35,41)(43,55)(44,78)(49,72)(53,56)(57,73)(58,74)(59,61)(60,75)(64,76)(66,67)(68,70)(69,77)(79,80)

Polycyclic presentation:

2^12|a^2=f,c^2=fh,f^2=h,g^2=i,h^2=j,i^2=k,k^2=l,b^a=bg,c^b=cikl,d^a=dj,d^c=dj,e^d=el,f^b=fkl,g^a=gi,g^b=gikl,g^c=gk,g^f=gl,h^b=hl,i^a=ik,i^b=ikl,i^c=il,k^a=kl,k^b=kl

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

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