#72320738

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	#72320473
commutator	C₂²×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	C₂×#2221	C₂×C₄
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ #479908 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

966 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:

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

Polycyclic presentation:

2^11|a^2=hjk,b^2=e,c^2=eh,d^2=fgj,e^2=h,f^2=i,i^2=j,j^2=k,b^a=bg,d^a=dghk,d^b=df,d^c=di,e^d=ejk,f^a=fk,f^b=fi,f^c=fj,f^d=fk,f^e=fk,g^a=gk,g^d=gk,h^d=hk,i^b=ij,i^c=ik,j^b=jk

Conjugacy classes & character table

113 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.8*F.10*F.11);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.5*F.8);;
gap> SetPower(coll, 4, F.6*F.7*F.10);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*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);