#72314363

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	#7622966	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂×#1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2915 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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=fhj,e^2=fgij,g^2=i,i^2=j,c^b=ch,d^a=dhj,d^c=dj,e^a=ej,e^b=ehj,e^c=ei,e^d=eg,f^e=fj,g^c=gj,g^d=gi,g^e=gj,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314371, #72314408, #72314559
maximal normal subgroup of	#72314371, #72314408, #72314559

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