#72309752

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21452
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#488891	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ D₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1918 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=j,c^2=gj,d^2=ej,f^2=h,g^2=i,i^2=j,c^a=cg,c^b=cf,d^c=dij,e^c=ej,f^b=fh,f^c=fh,g^a=gi,g^d=gj,i^a=ij

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309772, #72309828, #72310150
maximal normal subgroup of	#72309772, #72309828, #72310150

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