#72310796

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#107114
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂³
prime radical	C₂×#3500	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

2784 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)(3,13,15,40)(4,18,19,49)(6,26,27,35)(7,11,24,36)(8,29,31,17)(9,23,33,20)(12,25,38,28)(14,39,42,60)(16,41,22,53)(21,34,52,59)(30,54,55,62)(32,46,57,47)(37,50,58,51)(43,48,61,64)(44,56,45,63);(1,3)(2,8,11,32)(4,14)(5,22)(6,17)(7,16)(9,30,18,48)(10,31,36,57)(12,13,26,41)(15,24)(19,45)(20,44)(21,43)(23,42)(25,29)(27,46)(28,47)(33,55,49,64)(34,56,37,39)(35,53,38,40)(50,61)(51,62)(52,54)(58,60,59,63);(1,4)(2,9)(3,14)(5,23,24,19)(6,21)(7,20)(8,30)(10,33,36,49)(11,18)(12,37)(13,39)(15,45,22,42)(16,44)(17,43)(25,52,27,50)(26,34)(28,51)(29,54,46,61)(31,55,57,64)(32,48)(35,59,38,58)(40,60,53,63)(41,56)(47,62)

Polycyclic presentation:

2^10|a^2=d,b^2=hj,c^2=g,f^2=j,b^a=bf,c^a=ce,d^b=dh,d^c=dhi,e^a=ehi,e^b=eh,e^c=eh,e^d=ej,f^a=fhj,f^c=fhj,f^d=fj,f^e=fj,g^a=gh,g^d=gj,h^a=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310807, #72311042
maximal normal subgroup of	#72310807, #72311042

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