#72309786

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#488893	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8202 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1234 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=j,c^2=ehj,d^2=egi,f^2=h,g^2=i,i^2=j,c^a=chj,c^b=cf,d^a=dg,d^c=dhi,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

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309802, #72309999, #72310180
maximal normal subgroup of	#72309802, #72309999, #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.8*F.10);;
gap> SetPower(coll, 4, F.5*F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
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.8*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);