#72314054

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#33520	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=f,d^2=fg,e^2=ghj,h^2=i,i^2=j,c^a=cg,d^b=dgj,d^c=dg,e^a=eg,e^b=eg,e^c=eh,e^d=egi,f^e=fj,h^c=hi,h^d=hj,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314098, #72314160, #72314619
maximal normal subgroup of	#72314098, #72314160, #72314619

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