#72227022

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	#72227021
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₄×#346	C₂³
prime radical	C₂×#12436	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #493217 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

957 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:

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

Polycyclic presentation:

2^11|a^2=f,b^2=d,d^2=g,e^2=hjk,f^2=ijk,h^2=jk,j^2=k,b^a=be,c^a=cfhjk,c^b=cefik,d^a=dj,d^c=di,e^a=eh,e^b=ehjk,e^c=ehk,e^d=ek,f^b=fk,f^c=fij,g^a=gk,h^a=hj,h^b=hjk,h^c=hj,i^a=ik,i^b=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227037, #72227168, #72227179
maximal normal subgroup of	#72227037, #72227168, #72227179

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