#72260463

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	#513009	C₂
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

568 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=egh,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^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, #72322300
maximal normal subgroup of	#72260475, #72322276, #72322300

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.7*F.8);;
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, 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);