#72309859

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#22858
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	D₄×#135	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8202 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=j,c^2=ej,d^2=egij,f^2=h,g^2=i,i^2=j,b^a=bj,c^a=ch,c^b=cf,d^a=dg,d^c=di,e^d=ej,f^b=fh,f^c=fh,g^a=gi,g^c=gj,g^d=gj,i^a=ij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309870, #72309941, #72310180
maximal normal subgroup of	#72309870, #72309941, #72310180

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