#500885

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
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×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

748 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)(9,10,12,17)(11,21,23,15)(13,16,26,32)(14,19,27,20)(18,34,33,28)(22,30,25,35)(24,39,38,31)(29,37,36,40);(1,3,5,4)(2,6,8,7)(9,11,12,23)(10,15,17,21)(13,24,26,38)(14,22,27,25)(16,31,32,39)(18,29,33,36)(19,35,20,30)(28,37,34,40);(10,16)(11,22)(13,28)(14,27)(15,29)(17,32)(18,20)(19,33)(21,36)(23,25)(24,38)(26,34)(30,39)(31,35);(1,4,5,3)(2,7,8,6)(10,17)(15,21)(16,32)(18,33)(19,20)(29,36)(30,35)(31,39)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,g^b=gi,f^c=fh,g^c=gi,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, 408081);