#500751

order	512
nilpotency class	4
isoclinism	isoclinic class of #500749
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24248
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1003 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407947);