#72303852

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32898
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#8387757	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=hj,c^2=hj,d^2=fhij,e^2=j,c^a=chij,c^b=ci,d^a=dg,d^b=df,d^c=de,e^a=eh,e^b=ehi,e^d=ehi,f^c=fhij,g^a=gj,g^c=ghj,g^e=gj,h^d=hj,i^d=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72303861

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