#72224998

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	#500441
commutator, Frattini, cosocle	C₂²×C₁₆	C₂⁴
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ #33433 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1030 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=g,f^2=hi,h^2=ij,i^2=j,b^a=bfj,c^a=cij,d^a=de,d^b=dg,d^c=dg,e^a=ej,e^c=ej,f^b=fh,f^c=fj,f^d=fj,g^a=gj,h^b=hi,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225019
maximal normal subgroup of	#72225019

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