#72308255

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3205 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)(3,5)(4,6)(7,8)(10,15,16,26)(11,20,21,34)(13,17)(14,25)(18,22)(19,32,31,28)(23,38,37,33)(24,29)(27,40)(30,36);(1,3)(2,5)(4,7)(6,8)(10,16)(11,21)(15,26)(19,31)(20,34)(23,37)(28,32)(33,38);(3,7)(5,8)(9,10,17,32,25,31,13,26,12,16,29,28,14,19,24,15)(11,22,38,40,37,18,34,39,21,36,33,27,23,30,20,35);(11,21)(18,30)(20,34)(22,36)(23,37)(27,40)(33,38)(35,39);(2,6)(3,7)(9,11,13,20,14,23,17,33,12,21,24,34,25,37,29,38)(10,18,15,27,19,22,28,39,16,30,26,40,31,36,32,35)

Polycyclic presentation:

2^10|a^2=f,c^2=fhi,e^2=h,h^2=i,i^2=j,c^a=ch,c^b=cg,e^a=eghj,e^b=eg,e^c=ei,e^d=ej,f^c=fj,f^e=fj,h^a=hi,h^c=hj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308279, #72308337, #72308553
maximal normal subgroup of	#72308279, #72308337, #72308553

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