#72321533

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#486293
commutator, Frattini, cosocle	C₂×M₃₂	C₂⁴
LCS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ #33434 ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1203 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:

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

Polycyclic presentation:

2^10|a^2=e,c^2=g,e^2=hi,f^2=hi,h^2=ij,i^2=j,b^a=bej,c^a=cefh,d^a=dg,d^b=dg,d^c=dg,e^b=eh,e^c=eg,e^d=ej,f^a=fgj,f^b=fghj,f^c=fg,f^e=fj,g^a=gj,h^b=hi,i^b=ij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342904, #72342911, #72342980
maximal normal subgroup of	#72342904, #72342911, #72342980

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