#72250052

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	#10583437	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

3698 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=hj,c^2=hj,d^2=efi,e^2=j,f^2=ij,g^2=j,i^2=j,b^a=bj,c^a=cj,d^a=de,d^b=defi,d^c=degj,e^a=ej,e^b=eh,e^c=eh,e^d=eh,f^a=fh,f^b=fhij,f^c=fhj,f^d=fh,f^e=fj,g^a=gh,g^b=ghj,g^c=gh,g^d=gh,g^e=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250237
maximal normal subgroup of	#72250237
maximal quotient of	#72250539

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