#72313932

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	#7623353	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #20476 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3016 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)(3,7)(4,6)(5,8)(11,19)(16,30)(20,37)(21,38)(22,39)(23,32)(33,35)(34,40);(1,2,4,6)(3,5,8,7)(9,10,12,15)(11,20)(13,26,24,36)(14,29,25,18)(16,32)(17,27,31,28)(19,37)(21,40)(22,35)(23,30)(33,39)(34,38);(1,3,4,8)(2,5,6,7)(10,15)(11,21,19,38)(13,24)(16,33,30,35)(18,29)(20,34,37,40)(22,23,39,32)(27,28);(1,2)(3,5)(4,6)(7,8)(11,22)(13,27)(14,25)(16,34)(17,36)(18,29)(19,39)(20,35)(21,23)(24,28)(26,31)(30,40)(32,38)(33,37);(2,6)(5,7)(9,11,13,23,14,21,28,22,12,19,24,32,25,38,27,39)(10,16,17,20,18,35,26,34,15,30,31,37,29,33,36,40)

Polycyclic presentation:

2^10|b^2=f,c^2=fj,e^2=gh,g^2=i,i^2=j,b^a=bh,c^a=ch,c^b=cj,d^b=dh,e^a=ehj,e^b=eg,e^c=ei,e^d=egi,f^e=fj,g^b=gi,g^c=gj,g^d=gij,i^b=ij,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314059, #72314329, #72314594
maximal normal subgroup of	#72314059, #72314329, #72314594

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