#72309999

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#442
commutator	C₄×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	#72309991	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ SD₁₆×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^11|a^2=f,b^2=k,c^2=i,d^2=fk,e^2=hk,g^2=i,h^2=j,j^2=k,c^a=cik,c^b=cg,e^a=eh,e^c=ei,e^d=ej,f^e=fk,g^b=gi,g^c=gi,h^a=hj,h^d=hk,j^a=jk

Conjugacy classes & character table

137 conjugacy classes — view the full character table.

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