#72315223

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×#62328	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#532 ◁ C₂×#2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5495 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,32)(20,33)(21,34)(22,29)(23,35)(24,30)(25,36)(26,37)(27,38)(28,31)(39,40);(2,5)(3,7)(10,15)(16,29)(17,30)(18,31)(32,37)(33,35)(34,40)(36,38);(9,11,12,20,14,22,24,23)(10,16,17,33,15,29,30,35)(13,26,25,21,28,19,27,39)(18,37,36,34,31,32,38,40);(1,3)(2,6)(4,7)(5,8)(10,15)(16,29)(17,30)(18,31)(32,37)(33,35)(34,40)(36,38);(3,7)(6,8)(11,19)(12,24)(13,27)(16,32)(17,30)(18,38)(20,21)(22,26)(23,39)(25,28)(29,37)(31,36)(33,34)(35,40)

Polycyclic presentation:

2^10|c^2=f,f^2=j,g^2=i,i^2=j,b^a=bh,d^a=dj,d^b=dhj,e^c=eg,e^d=ehj,f^e=fj,g^c=gi,g^e=gij,i^c=ij,i^e=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315233, #72315390, #72315974
maximal normal subgroup of	#72315233, #72315390, #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.6);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);