#72317773

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#546 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5951 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=f,c^2=h,d^2=j,g^2=j,b^a=bi,c^a=cfh,c^b=cg,d^a=df,d^b=dij,d^c=dij,e^d=efi,f^c=fj,g^a=gi,g^b=gj,g^c=gij,g^d=gj,h^a=hj,h^b=hi,h^d=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317907, #72318828, #72319361
maximal normal subgroup of	#72317907, #72318828, #72319361

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