#72250047

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346546
commutator, Frattini, cosocle	C₂×#131	C₂⁴
prime radical	#10584355	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3424 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=hj,c^2=hj,d^2=f,e^2=j,f^2=i,g^2=j,i^2=j,c^a=cj,d^a=de,d^b=df,d^c=degj,e^a=ej,e^c=eh,e^d=ej,f^b=fi,f^c=fj,g^a=gh,g^b=gj,g^c=gh,g^d=gj,g^e=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250237
maximal normal subgroup of	#72250237
maximal quotient of	#72250479, #72279770

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