#72281040

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346542
commutator, Frattini, cosocle	C₂×#131	C₂⁴
prime radical	#10583256	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3720 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)(6,26)(10,14)(11,37)(12,32)(15,45)(17,29)(19,48)(20,30)(21,54)(22,50)(24,57)(28,53)(31,40)(34,39)(35,41)(38,49)(43,51)(44,60)(46,64)(47,56)(52,63)(55,59);(1,2,7,9)(3,14,8,10)(4,19)(5,12,25,32)(6,27,26,36)(11,38)(13,17,33,29)(15,46)(16,41,23,35)(18,48)(20,51)(21,31)(22,52)(24,58,57,62)(28,34)(30,43)(37,49)(39,53)(40,54)(42,47,61,56)(44,59)(45,64)(50,63)(55,60);(1,3,7,8)(2,10,9,14)(4,20,22,45)(5,23,25,16)(6,17,26,29)(11,39,31,60)(12,41,32,35)(13,36,33,27)(15,18,30,50)(19,51,52,64)(21,55,38,53)(24,56,57,47)(28,54,59,49)(34,40,44,37)(42,58,61,62)(43,63,46,48);(1,4,8,30,7,22,3,15)(2,11,14,44,9,31,10,34)(5,18,23,20,25,50,16,45)(6,28,29,21,26,59,17,38)(12,40,41,39,32,37,35,60)(13,43,27,19,33,46,36,52)(24,55,56,49,57,53,47,54)(42,51,58,48,61,64,62,63)

Polycyclic presentation:

2^10|b^2=h,c^2=j,d^2=hij,f^2=i,i^2=j,b^a=bh,c^b=ch,d^a=de,d^b=df,d^c=deg,e^c=eh,f^a=fj,f^b=fi,f^c=fhj,f^d=fi,g^a=ghj,g^b=ghj,g^c=ghj,g^d=gj,g^e=gj,g^f=gj,h^d=hj,i^b=ij,i^d=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.8*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, 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.9);;
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> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);