#72303186

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#106766
commutator	C₂²×Q₈	C₂×C₄²
Frattini, cosocle	C₂²×2₋¹⁺⁴	C₂³
prime radical	C₂×#3500	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×2₋¹⁺⁴ ▷ C₂³ ▷ C₂ ▷ C₁

2947 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)(3,13,7,9)(4,18,22,41)(5,11)(6,12)(8,14)(10,15)(16,37,24,30)(17,42,29,31)(19,47)(20,46,52,64)(21,53,55,32)(23,36)(25,38)(26,39)(27,40)(28,35)(33,43)(34,49)(44,58,45,57)(48,61)(50,63)(51,59,54,60)(56,62);(1,3)(2,9)(4,19,22,49)(5,16)(6,17)(7,8)(10,32,35,53)(11,30)(12,31)(13,14)(15,21,28,55)(18,34,41,47)(20,50,52,48)(23,44)(24,25)(26,45)(27,29)(33,60,62,59)(36,57)(37,38)(39,58)(40,42)(43,51,56,54)(46,61,64,63);(1,4,5,20,8,22,25,52)(2,10,11,33,14,35,38,62)(3,15,16,43,7,28,24,56)(6,21,26,54,27,55,23,51)(9,18,30,46,13,41,37,64)(12,34,39,63,40,47,36,61)(17,19,45,48,29,49,44,50)(31,32,58,59,42,53,57,60)

Polycyclic presentation:

2^10|a^2=g,b^2=h,c^2=d,d^2=j,e^2=j,f^2=j,b^a=bi,c^a=cf,c^b=ce,d^b=dh,e^a=eh,e^c=ehj,e^d=ej,f^a=fi,f^b=fhj,f^c=fj,f^e=fj,g^b=gj,g^c=gij,g^f=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303202
maximal normal subgroup of	#72303202

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