#502341

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	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

522 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)(3,7)(4,5)(6,8)(9,10,13,19,29,37,28,35,12,18,27,36,15,20,14,17)(11,21,24,38,40,33,25,34,23,16,22,31,26,39,30,32);(2,5)(6,7)(9,11,13,24,29,40,28,25,12,23,27,22,15,26,14,30)(10,16,19,31,37,39,35,32,18,21,36,38,20,33,17,34);(10,17)(11,22)(13,14)(15,29)(16,21)(18,35)(19,20)(23,24)(25,40)(26,30)(27,28)(31,32)(34,38)(36,37);(1,3,4,8)(2,6,5,7)(10,18)(16,21)(17,35)(19,36)(20,37)(31,38)(32,34)(33,39)

Polycyclic presentation:

2^9|a^2=f,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, 409537);