#72303183

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#394174
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
LCS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5012 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=h,d^2=f,b^a=bh,c^a=cij,c^b=ce,d^b=df,d^c=deg,e^a=ei,e^b=ei,e^d=ej,f^c=fj,g^a=gi,g^b=gi,h^c=hi,h^e=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303195, #72303348, #72303440
maximal normal subgroup of	#72303195, #72303348, #72303440

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