#72237310

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#333
commutator	C₄²	C₈²
Frattini, cosocle	C₄⁴	C₂²
socle	C₂²	#7025
prime radical	C₂⁴	#342
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₄⁴ ◁ G
LECS	G ▷ C₄⁴ ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1105 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 24 points:

(1,2,3,5,4,6,7,8)(9,10)(11,18,19,23)(12,15)(13,16)(14,20,22,24)(17,21);(9,11)(10,14,15,18,16,22,17,23)(12,20)(13,19)(21,24)

Polycyclic presentation:

2^10|a^2=c,b^2=d,c^2=f,d^2=g,e^2=ij,h^2=j,b^a=be,c^b=cj,d^a=dh,e^a=ei,e^b=ehij,g^a=gj,h^a=hj,i^b=ij

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72237314

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