#72260425

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁵
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ C₁₆×#41 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,d^2=fhj,e^2=g,f^2=hi,h^2=ij,i^2=j,c^a=cg,c^b=cg,d^a=df,d^b=dj,d^c=dj,e^d=ej,f^a=fh,h^a=hi,i^a=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260437
maximal normal subgroup of	#72260437

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