#72313746

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	#7623723	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ D₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2709 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=g,c^2=fi,d^2=hj,e^2=fgij,h^2=i,i^2=j,b^a=bj,c^a=ch,c^b=cg,d^a=dh,d^b=dj,d^c=dhj,e^a=ei,e^b=eg,e^d=egij,f^c=fj,f^d=fj,h^a=hi,h^c=hi,h^e=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313816, #72314441, #72314475
maximal normal subgroup of	#72313816, #72314441, #72314475

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