#72314031

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	#7623715	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ D₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|b^2=gj,c^2=f,d^2=f,e^2=fghi,h^2=i,i^2=j,b^a=bg,c^a=cg,c^b=cg,d^a=dg,e^c=eh,e^d=ei,f^e=fj,h^c=hi,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314098, #72314521, #72314559
maximal normal subgroup of	#72314098, #72314521, #72314559

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