#72227089

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	#7623661	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

2530 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=ej,d^2=g,e^2=h,f^2=ik,g^2=k,i^2=j,j^2=k,b^a=bf,c^a=ci,c^b=cfjk,d^c=dg,e^b=ej,e^c=ek,f^a=fi,f^b=fij,f^c=fi,f^e=fk,g^c=gk,h^b=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	#72227091, #72227158, #72227168
maximal normal subgroup of	#72227091, #72227158, #72227168

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