#72314480

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	#7623307	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂×#1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2883 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=fi,e^2=fh,g^2=i,i^2=j,c^a=cg,c^b=ch,d^b=dhj,d^c=dgj,e^a=ehj,e^b=ehj,e^c=eij,f^c=fj,g^a=gi,g^c=gi,g^d=gij,g^e=gj,i^a=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314497, #72314521, #72314559
maximal normal subgroup of	#72314497, #72314521, #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, 1, F.6);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 5, F.6*F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);