#72234337

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	C₂×#2177
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	#13663	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

562 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=eg,c^2=eg,d^2=f,e^2=gi,f^2=h,g^2=ij,i^2=j,b^a=be,c^a=ceh,c^b=chj,d^b=di,d^c=di,e^a=eg,e^d=ej,f^b=fj,f^c=fj,g^a=gi,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234355, #72234391, #72260475, …
maximal normal subgroup of	#72234355, #72234391, #72260475, …

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