#72339908

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 13634 conjugacy classes of subgroups; only the 39384 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)(10,13)(14,15);(9,10,12,13)(11,14,16,15);(2,4)(6,8)(17,18,20,23)(19,22,24,21);(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,24)(22,23);(6,8);(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,23)(22,24)

Polycyclic presentation:

2^11|b^2=i,c^2=k,b^a=bi,d^c=dk,g^a=gh,g^b=gi,g^c=gj,g^d=gi,g^e=ghijk,g^f=gik

Conjugacy classes & character table

308 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72339945, #72339997, #72340079

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