#511993

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,7,8,5)(10,16)(11,20)(12,23)(13,14)(17,31)(18,35)(19,30)(21,37)(22,27)(24,28)(25,26)(29,32)(33,40)(34,38);(1,3,4,8)(2,5,6,7);(9,10)(11,17)(12,24)(13,18)(14,28)(15,19)(16,23)(20,32)(21,38)(22,34)(25,30)(26,35)(27,33)(29,36)(31,37)(39,40);(9,11,15,22)(10,17,19,34)(12,21,26,37)(13,27,14,20)(16,29,30,40)(18,33,28,32)(23,36,25,39)(24,38,35,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419189);