#72251677

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#10725
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₂²×#117	C₂³
prime radical	C₂²×#535	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#535 ◁ G
LECS	G ▷ C₂²×#117 ▷ C₂³ ▷ C₂ ▷ C₁

3494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=e,d^2=i,e^2=j,g^2=j,b^a=bd,c^a=chj,c^b=cg,d^a=dj,d^b=dh,d^c=di,e^a=ehi,e^c=ej,e^d=ej,f^b=fij,g^a=gij,g^c=gj,h^b=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251679, #72251683, #72251724, …
maximal normal subgroup of	#72251679, #72251683, #72251724, …

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