#72286802

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#501097
commutator, Frattini, cosocle	C₂×#124	C₂⁴
LCS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1675 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:

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

Polycyclic presentation:

2^10|b^2=g,e^2=ij,h^2=j,b^a=bg,c^a=ce,c^b=ce,d^a=df,d^b=dij,d^c=dij,e^a=eij,e^b=eh,e^c=eij,e^d=ej,f^b=fj,f^c=fj,g^c=ghi,g^d=gj,g^e=gi,h^a=hi,h^b=hij,h^c=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286867
maximal normal subgroup of	#72286867

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