#72234389

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	#13697	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

517 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fhj,c^2=fh,d^2=egi,e^2=gij,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,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, #72260494, #72322276, …
maximal normal subgroup of	#72234391, #72260494, #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.6);;
gap> SetPower(coll, 2, F.6*F.8*F.10);;
gap> SetPower(coll, 3, F.6*F.8);;
gap> SetPower(coll, 4, F.5*F.7*F.9);;
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, 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, 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);