#72322534

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#442
commutator	C₂²×C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₄×C₈	C₂⁴
prime radical	C₂³×D₁₆	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ C₂×C₄×#322 ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 40 points:

(5,6)(7,8)(9,10,13,16,14,17,21,12)(11,15,20,23,18,22,24,19)(25,26,28,32,29,34,37,35)(27,31,38,33,39,30,36,40);(1,2,4,3)(6,8)(9,11,14,18)(10,15,17,22)(12,19,16,23)(13,20,21,24)(26,30)(27,36)(28,37)(31,34)(32,33)(35,40)(38,39);(1,3,4,2)(6,8)(11,18)(15,22)(19,23)(20,24);(9,12,21,17,14,16,13,10)(11,19,24,22,18,23,20,15)(25,27,37,36,29,39,28,38)(26,31,35,40,34,30,32,33)

Polycyclic presentation:

2^11|a^2=e,b^2=f,c^2=fik,d^2=ei,e^2=i,g^2=j,h^2=j,j^2=k,b^a=bg,c^a=cghjk,c^b=cik,d^b=djk,e^b=ek,g^a=gj,g^b=gjk,g^d=gk,h^a=hj,h^b=hjk,h^d=hk,j^a=jk,j^b=jk

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322727
maximal normal subgroup of	#72322727

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