#72268586

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

2743754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^11|a^2=h,g^2=i,c^a=cj,f^a=fh,g^a=gj,g^b=gjk,g^c=gk,g^d=gh,g^e=ghi,g^f=gi

Conjugacy classes & character table

344 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268601
maximal normal subgroup of	#72268601

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