#502397

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

Polycyclic presentation:

2^9|a^2=efi,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, 409593);