#72306336

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#128033
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂×#535	C₂³
prime radical	C₂×#3500	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ C₂×#3500 ◁ G
LECS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2226 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)(3,14)(4,20,24,38,25,56,30,62)(5,26,27,10)(6,29)(7,12)(8,19)(9,23,39,61,41,22,40,60)(11,31,43,28)(13,32)(15,47,21,57,49,63,53,55)(16,34,17,45)(18,36)(33,51,37,58,44,48,64,59)(35,52,42,50)(46,54);(1,3)(2,8)(4,21,25,53)(5,16)(6,18)(7,19)(9,37,41,64)(10,34)(11,42)(12,14)(13,36)(15,24,49,30)(17,27)(20,55,56,57)(22,58,23,59)(26,45)(28,50)(29,46)(31,52)(32,54)(33,39,44,40)(35,43)(38,47,62,63)(48,61,51,60);(1,4,5,22,7,25,27,23)(2,9,10,38,12,41,26,62)(3,15,16,48,19,49,17,51)(6,30,28,60,32,24,31,61)(8,33,34,57,14,44,45,55)(11,20,29,40,43,56,13,39)(18,53,50,59,54,21,52,58)(35,63,36,37,42,47,46,64)

Polycyclic presentation:

2^10|a^2=f,b^2=ij,c^2=d,d^2=j,f^2=ij,h^2=j,b^a=bj,c^a=ce,c^b=cghij,d^a=dg,e^a=eghj,e^b=ei,e^c=eg,e^d=ej,f^c=fgh,f^d=fj,f^e=fi,g^a=gj,g^c=gj,h^a=hi,h^c=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306348, #72306355, #72306389
maximal normal subgroup of	#72306348, #72306355, #72306389

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