#502398

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	#28240	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=ei,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, 409594);