#501578

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

501 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)(9,10,16,29,28,12,25,35,15,20,13,18,14,19,31,27)(11,17,30,40,22,26,38,36,24,34,21,32,23,33,37,39);(2,5)(6,7)(9,11)(10,17)(12,26)(13,21)(14,23)(15,24)(16,30)(18,32)(19,33)(20,34)(22,28)(25,38)(27,39)(29,40)(31,37)(35,36);(9,12,15,19)(10,14,20,28)(11,17,24,34)(13,18,16,29)(21,36,30,39)(22,33,23,26)(25,27,31,35)(32,37,40,38);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=efi,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=gi,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, 408774);