#72281740

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#121510
commutator	C₂×#131	C₂²×C₄
Frattini, cosocle	C₂²×#131	C₂³
prime radical	C₂²×#41	C₂×D₄
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

1124 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=g,c^2=ej,e^2=i,f^2=ij,h^2=j,i^2=j,b^a=bg,c^a=ce,c^b=cefi,d^c=dj,e^a=ei,e^b=ej,f^a=fh,f^b=fhij,g^c=ghij,g^f=gj,h^a=hj,h^b=hj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281861, #72282419
maximal normal subgroup of	#72281861, #72282419

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