#72329232

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

68571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^10|e^2=hi,f^2=hi,b^a=bg,e^a=egh,e^c=ej,e^d=egi,f^a=fgh,f^c=fj,f^d=fij,f^e=fgj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329238, #72339917, #72339970
maximal normal subgroup of	#72329238, #72339917, #72339970
maximal quotient of	#72336123, #72339937, #72340049

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