#72260465

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 ◁ #1450 ◁ #11399 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=efghij,d^2=f,e^2=gi,f^2=h,g^2=ij,i^2=j,c^a=ce,c^b=chij,d^b=dj,d^c=di,e^a=eg,e^b=ej,e^c=ej,e^d=ej,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	#72260475, #72322276
maximal normal subgroup of	#72260475, #72322276

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