#503301

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
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×#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

704 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,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,d^c=di,g^c=gi

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, 410497);