#72227047

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72216590
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	#512928	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #129051 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=k,b^2=d,c^2=dfjk,d^2=g,e^2=hj,f^2=ik,h^2=jk,j^2=k,b^a=be,c^a=chjk,c^b=cefik,d^a=djk,d^c=di,e^a=eh,e^b=ehjk,e^c=ehjk,e^d=ek,f^a=fj,f^b=fj,f^c=fik,g^a=gk,h^a=hj,h^b=hjk,h^c=hjk,i^a=ik,i^b=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227066, #72227138, #72227167
maximal normal subgroup of	#72227066, #72227138, #72227167

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