#72342763

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#1304
commutator	C₂⁴	C₄³
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	C₄×#1304
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

2663 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=f,c^2=g,d^2=j,c^b=ce,e^b=eh,e^c=ehi,f^c=fh,f^e=fj,g^b=ghi,g^e=gj,h^b=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342789, #72342855, #72342859
maximal normal subgroup of	#72342789, #72342855, #72342859

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