#72234390

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	#13661	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

552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 80 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fhj,c^2=fh,d^2=efgh,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,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, #72260494, #72322300, …
maximal normal subgroup of	#72234391, #72260494, #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.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.6*F.7*F.8);;
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, 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);