#500936

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

724 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408132);