#72262167

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 8256 conjugacy classes of subgroups; only the 34992 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:

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

Polycyclic presentation:

2^11|a^2=h,g^2=j,c^b=chj,d^a=dh,e^b=ei,f^c=fijk,f^e=fk,g^d=gj

Conjugacy classes & character table

266 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262179
maximal normal subgroup of	#72262179

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