#72250787

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346522
commutator, Frattini, cosocle	C₂×#131	C₂⁴
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3592 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)(3,15,21,36)(4,22)(5,27)(6,13)(7,10)(8,14)(9,19,37,20)(11,23)(12,29)(16,43,28,46)(17,41)(18,45,24,44)(25,38)(26,42)(30,33)(31,39)(32,50,40,49)(34,53,35,54)(47,62,48,61)(51,58)(52,60)(55,56)(57,64,59,63);(3,16,21,28)(4,23)(5,17)(6,8)(9,32,37,40)(11,22)(12,33)(13,14)(15,43,36,46)(18,47,24,48)(19,50,20,49)(25,52)(27,41)(29,30)(34,57,35,59)(38,60)(44,61,45,62)(51,56)(53,64,54,63)(55,58);(1,3,17,49,6,19,5,28,7,21,30,50,8,20,29,16)(2,9,33,46,13,15,12,40,10,37,41,43,14,36,27,32)(4,24,51,64,31,53,25,48,23,18,52,63,26,54,56,47)(11,34,58,61,42,44,38,57,22,35,60,62,39,45,55,59);(2,10)(3,18)(5,29)(9,34)(11,22)(13,14)(15,44)(16,47)(17,30)(19,54)(20,53)(21,24)(25,56)(28,48)(32,57)(35,37)(36,45)(39,42)(40,59)(43,61)(46,62)(49,64)(50,63)(51,52)

Polycyclic presentation:

2^10|a^2=hj,b^2=hj,c^2=fhij,f^2=i,g^2=j,i^2=j,c^a=cg,c^b=cf,d^a=dj,d^c=de,e^a=eh,e^b=ej,e^c=ej,f^a=fj,f^b=fi,f^c=fj,f^d=fj,g^b=gj,g^d=ghj,g^e=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250830
maximal normal subgroup of	#72250830
maximal quotient of	#72250902

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