#503136

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1150 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410332);