#72234349

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#367
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂×#41	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ C₁₆×#41 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 80 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=j,d^2=egi,e^2=gij,f^2=h,g^2=ij,i^2=j,c^a=ch,c^b=ch,d^a=de,d^c=dj,e^a=eg,f^d=fj,g^a=gi,i^a=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234355
maximal normal subgroup of	#72234355

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