#72267800

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#392
commutator, Frattini, cosocle	C₂³×C₄²	C₂³
prime radical	C₂×#2856	C₂²
LCS	G ▷ C₂³×C₄² ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂⁴ ▷ C₁

2850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=hi,b^2=i,c^2=ej,d^2=g,f^2=ij,b^a=bd,c^a=ce,c^b=cefij,d^a=dg,d^b=dg,d^c=dh,e^b=ei,f^a=fhi,f^b=fj,f^c=fj

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267804
maximal normal subgroup of	#72267804
maximal quotient of	#72267821

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