#501769

order	512
nilpotency class	4
isoclinism	isoclinic class of #501768
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24836
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

515 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408965);