#72314539

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	#7623725	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ D₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2672 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)(3,8)(4,6)(5,7)(9,10)(11,19)(12,17)(13,15)(14,18)(16,24)(20,31)(21,29)(22,33)(23,27)(25,32)(26,28)(30,37)(34,40)(35,39)(36,38);(1,2)(3,6)(4,8)(5,7)(11,20,24,21)(12,23)(16,29,19,31)(17,27)(22,34,37,35)(25,36)(30,39,33,40)(32,38);(1,3,5,4)(2,6,7,8)(10,15)(11,21,24,20)(12,23)(16,29,19,31)(18,28)(22,35,37,34)(25,36)(30,39,33,40);(2,7)(6,8)(11,22,24,37)(12,25)(14,26)(16,30,19,33)(17,32)(18,28)(20,34,21,35)(23,36)(27,38)(29,39,31,40);(1,4,5,3)(2,8,7,6)(9,11,23,37,14,21,36,35,13,24,12,22,26,20,25,34)(10,16,27,33,18,31,38,40,15,19,17,30,28,29,32,39)

Polycyclic presentation:

2^10|b^2=f,c^2=fhj,d^2=f,e^2=gh,g^2=i,i^2=j,c^a=ch,d^a=dhj,d^b=dhj,e^a=eh,e^b=eij,e^c=ei,e^d=eg,f^e=fj,g^b=gj,g^c=gj,g^d=gi,i^d=ij

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314559, #72315244, #72315286
maximal normal subgroup of	#72314559, #72315244, #72315286

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