#509202

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	#28484	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

803 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)(9,10,14,19)(11,21,16,17)(12,25,13,27)(15,30,29,35)(18,28,36,26)(20,24,37,33)(22,40,23,38)(31,39,32,34);(1,2)(3,4)(5,6)(7,8)(9,11,14,16)(10,17,19,21)(12,22,13,23)(15,24,29,33)(18,34,36,39)(20,35,37,30)(25,38,27,40)(26,31,28,32);(6,8)(9,12,14,13)(10,18,19,36)(11,22,16,23)(15,31,29,32)(17,34,21,39)(20,38,37,40)(24,28,33,26)(25,35,27,30);(2,4)(9,13,14,12)(10,18,19,36)(11,23,16,22)(15,32,29,31)(17,34,21,39)(20,38,37,40)(24,26,33,28)(25,35,27,30)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^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, 416398);