#72297704

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 13018 conjugacy classes of subgroups; only the 29441 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 24 points:

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

Polycyclic presentation:

2^10|d^2=h,b^a=bj,c^b=ch,d^a=dh,d^b=dh,d^c=dh,e^a=ei,e^c=ei,e^d=eh,f^b=fh,f^d=fh,f^e=fh,g^b=gi,g^d=gi,g^e=gi,g^f=gj

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72297775, #72297910, #72298061

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