#72297583

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 43223 conjugacy classes of subgroups; only the 34804 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^11|d^2=i,b^a=bk,c^b=ci,d^a=dh,d^b=di,d^c=di,e^a=ehij,e^c=ej,e^d=ei,f^b=fi,f^d=fi,f^e=fi,g^b=ghij,g^d=ghij,g^e=ghij,g^f=gj

Conjugacy classes & character table

254 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72298631

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);