#490812

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	#33579	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

473 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)(9,10,16,21)(11,20,24,38)(12,25,28,32)(13,19,14,29)(15,30,23,17)(18,33,35,22)(26,40,27,37)(31,34,39,36);(1,2,3,5,4,6,7,8)(9,11,14,23,16,24,13,15)(10,17,19,20,21,30,29,38)(12,22,27,39,28,33,26,31)(18,32,34,37,35,25,36,40);(9,12,16,28)(10,18,21,35)(11,22,24,33)(13,26,14,27)(15,31,23,39)(17,32,30,25)(19,34,29,36)(20,37,38,40);(2,6)(5,8)(9,13,16,14)(10,19,21,29)(11,15,24,23)(12,26,28,27)(17,20,30,38)(18,34,35,36)(22,31,33,39)(25,40,32,37)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,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, 398008);