#72314011

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#62431	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ C₂×#2966 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5432 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:

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

Polycyclic presentation:

2^10|c^2=hj,e^2=f,f^2=ij,g^2=ij,i^2=j,b^a=bh,c^a=ch,c^b=chj,d^b=dh,e^a=ehj,e^b=ej,e^c=eg,e^d=egj,f^c=fi,f^d=fi,g^c=gi,g^d=gi,g^e=gj,i^c=ij,i^d=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314059, #72314098, #72315974
maximal normal subgroup of	#72314059, #72314098, #72315974

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