#72314543

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	#7623625	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ D₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3010 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)(9,10)(11,19)(12,17)(13,18)(14,15)(16,24)(20,37)(21,32)(22,38)(23,34)(25,36)(26,29)(27,31)(28,30)(33,39)(35,40);(1,3)(2,5)(4,8)(6,7)(11,20,24,27)(12,25)(13,28)(16,31,19,37)(17,36)(18,30)(21,39,40,22)(23,26)(29,34)(32,33,35,38);(2,6)(5,7)(10,15)(11,21,24,40)(12,26)(16,32,19,35)(18,30)(20,39,27,22)(23,25)(31,33,37,38);(2,6)(3,8)(11,22)(12,23)(13,28)(16,33)(17,34)(18,30)(19,38)(20,21)(24,39)(25,26)(27,40)(29,36)(31,35)(32,37);(2,6)(5,7)(9,11,23,22,28,40,26,20,14,24,25,39,13,21,12,27)(10,16,34,33,30,32,29,31,15,19,36,38,18,35,17,37)

Polycyclic presentation:

2^10|b^2=f,c^2=f,e^2=fgij,g^2=i,i^2=j,b^a=bh,c^a=chj,d^a=dh,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,g^e=gj,i^b=ij,i^d=ij

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314559, #72314604, #72315361
maximal normal subgroup of	#72314559, #72314604, #72315361

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, 5, F.6*F.7*F.9*F.10);;
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*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
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, 7, 5, F.7*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);