#72237898

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72235715
commutator, Frattini, cosocle	#2308	C₂⁴
prime radical	#72237830	C₂
LCS	G ▷ #2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁵ ◁ #35348 ◁ G
LECS	G ▷ #2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

4725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^11|c^2=ij,d^2=i,e^2=ik,f^2=ij,b^a=bg,c^b=cf,d^a=dhjk,d^b=dijk,d^c=de,e^a=ejk,e^b=eh,e^c=ei,e^d=eik,f^a=fij,f^b=fij,f^c=fij,f^d=fhjk,f^e=fj,g^c=gij,g^d=gk,g^e=gk,h^c=hj,h^d=hk,h^e=hk,h^f=hk,i^b=ik,j^b=jk,j^d=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237931
maximal normal subgroup of	#72237931

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