#72328045

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

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 12904 conjugacy classes of subgroups; only the 39184 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:

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

Polycyclic presentation:

2^11|f^2=i,g^2=hik,b^a=bh,c^b=chk,f^a=fhj,f^c=fk,f^d=fhj,f^e=fi,g^a=gjk,g^c=gk,g^d=gjk,g^e=ghik,g^f=ghk

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72335839

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