#72281156

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	#346518
commutator, Frattini, cosocle	C₂×#131	C₂⁴
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3972 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:

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

Polycyclic presentation:

2^10|b^2=h,c^2=j,d^2=efij,f^2=i,i^2=j,b^a=bh,c^b=ch,d^a=de,d^b=df,d^c=deg,e^c=eh,e^d=ej,f^a=fj,f^b=fi,f^c=fhj,f^d=fj,g^a=ghj,g^b=ghj,g^c=ghj,g^d=gj,g^e=gj,g^f=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281944, #72307865
maximal normal subgroup of	#72281944, #72307865

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