#72233898

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#10965
commutator	C₂⁴×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄×D₄	C₂³
prime radical	#649037	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#546 ◁ G
LECS	G ▷ C₂²×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=g,c^2=d,d^2=j,f^2=i,b^a=bg,c^a=cf,c^b=ce,d^a=dh,d^b=dhi,e^c=ehi,e^d=ej,f^b=fi,f^c=fhi,f^d=fj,g^c=gi,h^c=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233903, #72234088, #72234090
maximal normal subgroup of	#72233903, #72234088, #72234090

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