#502480

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

695 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)(9,10,15,21,13,30,27,39,14,20,23,33,16,19,32,38)(11,22,25,36,28,34,40,37,24,17,31,18,12,26,29,35);(1,2,4,3)(5,6)(7,8)(9,11,23,31,13,28,32,29,14,24,15,25,16,12,27,40)(10,17,33,36,30,26,38,37,20,22,21,18,19,34,39,35);(6,8)(9,12,14,28)(10,18,20,36)(11,13,24,16)(15,31,23,25)(17,33,22,21)(19,35,30,37)(26,39,34,38)(27,40,32,29);(1,3,4,2)(9,13,14,16)(10,19,20,30)(11,12,24,28)(15,27,23,32)(17,26,22,34)(18,35,36,37)(21,38,33,39)(25,29,31,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409676);