#72225012

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#500118
commutator, Frattini, cosocle	C₂²×C₁₆	C₂⁴
prime radical	#7623644	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ #33434 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1355 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:

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

Polycyclic presentation:

2^10|a^2=f,d^2=gi,e^2=j,f^2=hi,g^2=j,h^2=ij,i^2=j,b^a=bfgj,d^a=de,d^b=dgi,d^c=dgi,e^a=ej,e^b=ej,e^c=ej,f^b=fh,g^a=gj,g^b=gj,h^b=hi,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225019, #72320782, #72321336
maximal normal subgroup of	#72225019, #72320782, #72321336

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