#72306734

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#13346
commutator, Frattini, cosocle	#532	C₂⁴
prime radical	#7622903	C₂
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

3400 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=gh,e^2=j,g^2=i,h^2=j,b^a=bg,c^a=cij,c^b=ci,d^a=de,d^b=df,e^a=ej,e^b=eh,e^c=ej,e^d=ej,f^a=fij,f^b=fhij,f^c=fj,f^e=fj,g^a=gi,g^b=gi,g^d=ghi,g^e=gi,g^f=gi,h^a=hi,h^b=hi,h^e=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306739
maximal normal subgroup of	#72306739

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