#72234383

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

512 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:

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

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^b=ch,d^a=de,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	#72234391, #72322300, #72322532
maximal normal subgroup of	#72234391, #72322300, #72322532

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, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
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);