#72331244

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 35260 conjugacy classes of subgroups; only the 4964 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^11|c^2=i,e^2=g,f^2=ijk,b^a=bj,c^a=cjk,c^b=cg,d^b=di,d^c=di,e^a=ejk,e^b=eg,e^c=egi,e^d=ei,f^d=fijk,f^e=fgh

Conjugacy classes & character table

218 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331654
maximal normal subgroup of	#72331654

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