#495836

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₄×#459
prime radical	#33594	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

666 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=g,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403032);