#502396

order	512
nilpotency class	4
isoclinism	isoclinic class of #502133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409592);