#72225558

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#498306
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₂²×C₁₆	C₂⁴
prime radical	#512928	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ #12664 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

801 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=f,c^2=fghij,d^2=ghij,e^2=hi,g^2=ij,h^2=ij,i^2=j,b^a=be,c^a=ci,c^b=cegi,d^a=di,d^b=di,d^c=dghij,e^a=eh,e^b=ehj,e^c=ehj,f^a=fj,g^a=gij,g^b=gij,g^c=gi,h^a=hi,h^b=hi,h^c=hi,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225578, #72226878, #72226893
maximal normal subgroup of	#72225578, #72226878, #72226893

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