#72336142

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 22188 conjugacy classes of subgroups; only the 5868 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:

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

Polycyclic presentation:

2^11|a^2=g,b^2=gk,f^2=ghj,b^a=bh,c^a=cghj,c^b=cghk,d^a=di,d^b=di,e^a=egik,e^b=egik,f^a=fk,f^c=fghj,f^d=fhijk,f^e=fgik

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336150
maximal normal subgroup of	#72336150

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