#72320828

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72320588
commutator	C₂²×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	#153097	C₄
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ #479960 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1001 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

(2,8,12,31,13,27,14,33)(4,22,24,29)(5,25)(9,28,23,54,35,37,46,32)(10,36,39,40)(11,41,44,61,16,30,45,59)(18,50,47,57)(19,20)(26,49)(34,58,55,60,42,56,51,62)(38,63,43,64)(52,53);(2,9,14,46,13,35,12,23)(3,15)(4,10,24,39)(5,26)(6,17)(8,28,33,32,27,37,31,54)(11,42,45,55,16,34,44,51)(18,43,47,38)(19,52)(20,53)(22,36,29,40)(25,49)(30,58,61,62,41,56,59,60)(50,64,57,63);(1,2,10,37,49,33,4,23,17,12,36,54,25,8,29,9,7,13,39,28,26,31,24,46,6,14,40,32,5,27,22,35)(3,16,38,56,53,59,18,51,48,45,63,60,19,41,57,42,15,11,43,58,52,61,47,55,21,44,64,62,20,30,50,34);(1,3,17,48,7,15,6,21)(2,11,12,44,13,16,14,45)(4,18,29,57,24,47,22,50)(5,20,49,53,25,19,26,52)(8,30,31,59,27,41,33,61)(9,34,46,51,35,42,23,55)(10,38,36,63,39,43,40,64)(28,56,32,60,37,58,54,62)

Polycyclic presentation:

2^11|a^2=e,b^2=eh,c^2=efi,d^2=jk,e^2=h,f^2=i,g^2=j,i^2=j,j^2=k,c^a=cg,c^b=cf,d^a=dh,d^b=dgijk,d^c=dk,e^c=ej,f^a=fj,f^b=fi,f^c=fj,f^e=fk,g^a=gk,g^b=gj,g^c=gk,g^d=gk,h^c=hk,i^a=ik,i^b=ij,i^c=ik,j^b=jk

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321568
maximal normal subgroup of	#72321568

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