#72226370

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#115690
commutator	C₂⁴×D₄	C₂²×C₄
Frattini, cosocle	C₂³×H(5,𝔽₂)	C₂³
prime radical	#72225690	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×H(5,𝔽₂) ▷ C₂⁴ ▷ C₂ ▷ C₁

24922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^11|c^2=d,b^a=bg,c^a=cf,c^b=ce,d^a=dk,d^b=di,e^a=eh,e^c=ei,f^b=fhj,f^c=fk,f^e=fk,g^c=gjk,g^d=gk,i^a=ik,j^c=jk

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258565, #72265910, #72265932, …
maximal normal subgroup of	#72258565, #72265910, #72265932, …
maximal quotient of	#72226375, #72226387, #72266176, …

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