#509246

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416442);