#72234382

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 ◁ C₈×M₁₆ ◁ #11408 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

536 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=fj,d^2=efgij,e^2=gi,f^2=h,g^2=ij,i^2=j,c^b=ch,d^a=de,d^b=dij,d^c=dij,e^a=eg,e^b=ej,e^c=ej,e^d=ej,f^d=fj,g^a=gi,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234391, #72322276, #72322316
maximal normal subgroup of	#72234391, #72322276, #72322316

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