#72314569

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	#7623363	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ #20436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2983 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=f,d^2=hj,e^2=g,g^2=i,i^2=j,c^a=ch,c^b=ch,d^a=dj,d^b=dh,e^a=eh,e^b=ehij,e^c=eg,f^e=fj,g^b=gj,g^c=gi,i^c=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314574, #72314604, #72315244
maximal normal subgroup of	#72314574, #72314604, #72315244

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