#72320008

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#365433
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #10711 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2221 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:

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

Polycyclic presentation:

2^10|a^2=e,d^2=ef,e^2=ij,f^2=i,g^2=j,h^2=j,i^2=j,b^a=bf,c^a=cfgj,c^b=cf,d^a=dh,d^b=dj,d^c=di,e^b=ei,e^c=eij,e^d=ej,f^a=fj,f^b=fij,f^c=fij,f^d=fj,g^b=gj,g^c=gj,g^d=gj,h^b=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320043, #72320733, #72343980
maximal normal subgroup of	#72320043, #72320733, #72343980

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