#490816

order	512
nilpotency class	4
isoclinism	isoclinic class of #490801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

465 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,h^a=hi,d^b=dgi,f^c=fh,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398012);