#72314063

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	C₂³×#106
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#121	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ C₂×#2966 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2520 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)(19,34)(20,30)(21,26)(22,31)(23,32)(24,27)(25,28)(29,35)(33,36)(37,39)(38,40);(3,4)(6,8)(10,15)(11,19,21,35)(12,20)(13,25)(16,29,26,34)(17,33)(22,23,37,38)(24,36)(27,30)(31,40,39,32);(1,3)(2,6)(4,5)(7,8)(10,15)(16,26)(17,27)(18,28)(29,34)(30,33)(31,39)(32,40);(1,4,5,3)(2,6,7,8)(9,11,20,35,25,38,12,22,14,21,36,19,13,23,24,37)(10,16,30,29,28,40,17,31,15,26,33,34,18,32,27,39)

Polycyclic presentation:

2^10|b^2=g,c^2=f,e^2=ghi,h^2=i,i^2=j,c^b=cj,d^a=dg,d^b=dj,d^c=dg,e^a=eg,e^b=eg,e^c=eghij,e^d=eg,f^e=fj,h^c=hi,i^c=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314098, #72314251, #72315522
maximal normal subgroup of	#72314098, #72314251, #72315522

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