#503658

order	512
nilpotency class	4
isoclinism	isoclinic class of #503658
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25958
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3584	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

633 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)(3,17,23,54)(4,24)(5,29,18,55)(6,32)(7,15)(8,14)(9,16)(10,20,40,60)(11,19)(12,30,45,56)(13,36)(21,46,22,53)(25,51,31,47)(26,41)(27,43)(28,42)(33,50,59,39)(34,44,62,38)(35,49)(37,48)(52,61)(57,63)(58,64);(1,3)(2,10)(4,25)(5,19)(6,33)(7,21)(8,22)(9,23)(11,12)(13,46)(14,39)(15,50)(16,40)(17,48)(18,27)(20,35)(24,38)(26,56)(28,31)(29,52)(30,58)(32,54)(34,57)(36,60)(37,59)(41,47)(42,44)(43,45)(49,53)(51,64)(55,63)(61,62);(1,4,7,19,9,28,8,27)(2,11,14,24,16,43,15,42)(3,18,21,25,23,5,22,31)(6,26,35,57,37,58,36,61)(10,38,39,45,40,44,50,12)(13,41,48,52,49,64,32,63)(17,51,53,55,54,47,46,29)(20,56,59,34,60,30,33,62);(1,5)(2,12)(3,19)(4,22)(6,34)(7,31)(8,25)(9,18)(10,11)(13,47)(14,38)(15,44)(16,45)(17,52)(20,58)(21,28)(23,27)(24,39)(26,60)(29,48)(30,35)(32,55)(33,57)(36,56)(37,62)(40,43)(41,46)(42,50)(49,51)(53,64)(54,63)(59,61)

Polycyclic presentation:

2^9|a^2=fh,c^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,e^b=ehi,g^b=gi,h^b=hi,g^c=gi,e^d=ehi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410854);