#72227017

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#1333
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	C₂×#33858	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#443 ◁ #493217 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2590 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:

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

Polycyclic presentation:

2^11|a^2=k,b^2=e,d^2=gik,e^2=h,f^2=ij,g^2=jk,h^2=k,i^2=jk,j^2=k,b^a=bf,c^b=cfgj,d^c=dgi,e^a=ejk,e^c=ej,f^a=fi,f^b=fijk,f^c=fi,f^e=fk,g^a=gjk,g^b=gj,g^c=gj,h^a=hk,h^c=hk,i^a=ij,i^b=ijk,i^c=ij,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227037, #72227158, #72227166
maximal normal subgroup of	#72227037, #72227158, #72227166

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