#72281176

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	#106962
commutator	C₂²×C₈	C₂×C₄²
Frattini, cosocle	C₂²×#131	C₂³
prime radical	C₂²×#576	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2860 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)(3,14,17,35)(4,9,22,36)(5,10,27,40)(6,12)(7,13)(8,18)(11,28)(15,44,48,64)(16,51)(19,45)(20,47,55,46)(21,34)(23,37)(24,42,25,38)(26,39,29,43)(30,49)(31,53)(32,50,63,56)(33,58)(41,57)(52,62)(54,60)(59,61);(3,15)(5,26)(6,28)(8,30)(10,39)(11,12)(14,44)(16,23)(17,48)(18,49)(19,58)(20,56)(21,59)(24,25)(27,29)(31,62)(32,47)(33,45)(34,61)(35,64)(37,51)(38,42)(40,43)(41,60)(46,63)(50,55)(52,53)(54,57);(1,3,16,48,6,20,52,56,7,17,53,15,28,55,23,50)(2,8,31,30,12,33,37,60,13,34,51,61,11,41,62,45)(4,21,26,49,25,57,27,54,22,18,29,59,24,58,5,19)(9,35,43,64,38,46,10,32,36,14,39,44,42,47,40,63)

Polycyclic presentation:

2^10|a^2=d,c^2=defg,f^2=ij,g^2=j,i^2=j,c^a=ce,c^b=cefi,d^c=dh,e^a=eh,e^c=ej,e^d=ej,f^a=fh,f^b=fi,f^d=fj,g^a=gj,g^b=gj,g^c=ghj,g^e=gj,g^f=gj,h^a=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281197, #72281861, #72282517
maximal normal subgroup of	#72281197, #72281861, #72282517

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