#72235361

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72228242
commutator, Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂⁴
prime radical	#8934620	C₂²
LCS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

3503 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=efh,c^2=egjk,d^2=egjk,e^2=k,g^2=k,b^a=bef,c^a=chik,c^b=cg,d^a=dgj,d^b=dfhjk,d^c=degjk,e^b=ei,e^c=eij,e^d=ejk,f^a=fi,f^b=fijk,f^c=fijk,f^d=fj,f^e=fk,g^a=gj,g^b=gjk,g^c=gijk,g^d=gjk,g^f=gk,h^a=hijk,h^b=hjk,h^c=hjk,h^d=hi,h^e=hk,i^a=ik,i^c=ik,i^d=ik,j^b=jk,j^c=jk

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72235400

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