#491785

order	512
nilpotency class	4
isoclinism	isoclinic class of #491712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

469 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:

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

Polycyclic presentation:

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

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, 398981);