#509198

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

799 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:

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

Polycyclic presentation:

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

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, 416394);