#72308263

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	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	C₂²×#442
prime radical	C₂×#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3243 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=fgij,d^2=fghi,f^2=j,h^2=i,i^2=j,b^a=bg,c^a=cgi,d^a=dh,d^b=dg,d^c=dij,e^a=ej,f^d=fj,h^a=hi,h^c=hj,h^d=hj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308279, #72308363, #72308621
maximal normal subgroup of	#72308279, #72308363, #72308621

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