#509251

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
prime radical	#25805	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

701 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)(9,10,13,18)(11,22,25,16)(12,28,30,34)(14,17,32,37)(15,20,33,21)(19,24,38,29)(23,35,27,39)(26,40,31,36);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,29)(13,25)(14,26)(15,23)(17,36)(18,22)(19,34)(20,39)(21,35)(24,30)(27,33)(28,38)(31,32)(37,40);(10,17)(11,23)(14,29)(15,33)(16,34)(18,37)(19,21)(20,38)(22,28)(24,32)(25,27)(26,31)(35,40)(36,39);(2,6)(3,8)(9,12,13,30)(10,16,18,22)(11,24,25,29)(14,23,32,27)(15,26,33,31)(17,34,37,28)(19,36,38,40)(20,35,21,39)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,d^2=i,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, 416447);