#503300

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

714 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)(11,20)(12,26)(13,18)(14,29)(15,19)(16,24)(17,22)(21,34)(23,35)(25,36)(27,33)(28,39)(30,37)(31,38)(32,40);(1,3,5,4)(2,6,8,7)(9,10)(11,21)(12,17)(13,29)(14,18)(15,19)(16,25)(20,34)(22,26)(23,32)(24,36)(27,39)(28,33)(30,31)(35,40)(37,38);(9,11,22,34,14,25,28,38,15,23,27,40,13,30,12,24)(10,16,26,37,18,32,33,35,19,31,39,36,29,21,17,20);(1,4,5,3)(2,7,8,6)(11,23)(16,31)(20,35)(21,32)(24,38)(25,30)(34,40)(36,37)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=e,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, 410496);