#72309644

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1893
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂×#19258	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,5,8,7)(9,10,15,18)(11,16,14,30)(12,22,26,33)(13,27,29,40)(17,23,32,24)(19,21,34,28)(20,35,25,39)(31,36,38,37);(2,5)(3,8)(6,7)(9,11,15,14)(10,16,18,30)(12,20,26,25)(13,21,29,28)(17,31,32,38)(19,27,34,40)(22,35,33,39)(23,36,24,37);(9,12,13,23,11,20,21,36,15,26,29,24,14,25,28,37)(10,17,19,33,30,38,40,35,18,32,34,22,16,31,27,39);(2,6)(5,7)(10,18)(16,30)(17,32)(19,34)(22,33)(27,40)(31,38)(35,39)

Polycyclic presentation:

2^10|a^2=ij,b^2=j,c^2=e,e^2=hj,f^2=hj,g^2=i,h^2=j,b^a=bg,c^a=cf,d^a=dij,e^a=eh,f^a=fh,f^c=fj,g^a=gi,g^b=gi,h^a=hj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309663, #72310180
maximal normal subgroup of	#72309663, #72310180

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