#72342935

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#130718
commutator	C₂²×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×#504	C₂³
prime radical	#7623618	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ #33433 ◁ G
LECS	G ▷ C₂×#504 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1151 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:

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

Polycyclic presentation:

2^10|c^2=efh,e^2=hi,f^2=hi,h^2=ij,i^2=j,b^a=bg,c^a=ce,c^b=cd,d^c=dg,e^a=eh,e^b=ej,e^c=eh,f^a=fh,f^b=fgj,f^c=fh,f^d=fj,g^c=gj,h^a=hi,h^c=hi,i^a=ij,i^c=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342936, #72342960, #72343025
maximal normal subgroup of	#72342936, #72342960, #72343025

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.5*F.6*F.8);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);