#72306388

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#128073
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂×#535	C₂³
prime radical	C₂×#3449	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂×#3449 ◁ G
LECS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2449 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)(4,20,23,54,25,40,24,36)(5,26,27,10)(6,29)(7,12)(8,19)(9,22,37,56,38,57,53,58)(11,30,41,28)(13,31)(15,45,48,61,21,55,49,59)(16,33,17,43)(18,35)(32,46,60,63,42,47,62,64)(34,51,39,50)(44,52);(1,3)(2,8)(4,21)(5,16)(6,18)(7,19)(9,32)(10,33)(11,39)(12,14)(13,35)(15,25)(17,27)(20,45)(22,47)(23,49)(24,48)(26,43)(28,50)(29,44)(30,51)(31,52)(34,41)(36,59)(37,60)(38,42)(40,55)(46,57)(53,62)(54,61)(56,64)(58,63);(1,4,5,22)(2,9,10,36)(3,15,16,46)(6,24,28,56)(7,25,27,57)(8,32,33,59)(11,40,13,37)(12,38,26,54)(14,42,43,61)(17,47,19,21)(18,49,50,63)(20,29,53,41)(23,30,58,31)(34,45,44,62)(35,60,39,55)(48,51,64,52)

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306389, #72306394
maximal normal subgroup of	#72306389, #72306394

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, 3, F.4);;
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.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.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
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, 8, 1, F.8*F.9);;
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);