#72322567

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#442
commutator	C₂³×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₄×C₈	C₂⁴
prime radical	C₂³×D₁₆	C₂×C₄
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ C₂×#8260 ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4330 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:

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

Polycyclic presentation:

2^11|a^2=h,b^2=e,d^2=fgi,e^2=i,h^2=j,j^2=k,b^a=bhjk,c^a=ch,d^a=df,d^b=dfg,d^c=di,e^a=ek,h^b=hj,h^c=hjk,j^b=jk,j^c=jk

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322727, #72322777
maximal normal subgroup of	#72322727, #72322777

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