#72234351

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#357
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	#7623995	C₂
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

983 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=j,d^2=efgh,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,e^d=ej,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.6*F.7*F.8);;
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, 5, 4, F.5*F.10);;
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);