#72314458

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×#62270	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#525 ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5519 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)(4,8)(5,7)(11,19)(16,20)(21,22)(23,34)(29,30)(31,35)(36,37)(39,40);(1,3)(2,6)(4,5)(7,8)(9,10)(11,20)(12,17)(13,15)(14,18)(16,19)(21,29)(22,30)(23,35)(24,33)(25,28)(26,27)(31,34)(32,38)(36,39)(37,40);(1,2)(3,8)(4,6)(5,7)(10,15)(11,21)(12,24)(14,25)(16,29)(17,32)(19,22)(20,30)(23,37)(26,38)(27,33)(31,40)(34,36)(35,39);(1,4,5,3)(2,6,7,8)(10,15)(11,19)(17,27)(18,28)(21,22)(23,34)(32,33)(36,37);(2,7)(6,8)(9,11,12,22,25,34,38,36,13,19,26,21,14,23,24,37)(10,16,17,29,28,35,32,40,15,20,27,30,18,31,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314475, #72314574, #72315974
maximal normal subgroup of	#72314475, #72314574, #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, 4, 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, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
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.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);