#72286012

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	#72235717
commutator, Frattini, cosocle	#2309	C₂⁴
prime radical	#72284475	C₂
LCS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35272 ◁ G
LECS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

4971 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:

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

Polycyclic presentation:

2^11|b^2=gi,c^2=jk,d^2=i,e^2=k,g^2=i,b^a=bg,c^a=cf,c^b=cfhik,d^a=di,d^c=de,e^a=ei,e^b=eik,e^c=ek,f^b=fh,f^d=fi,f^e=fk,g^a=gi,g^c=ghj,g^f=gjk,h^a=hj,h^b=hjk,h^g=hk,i^c=ik,j^b=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286380
maximal normal subgroup of	#72286380

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