#491516

order	512
nilpotency class	4
isoclinism	isoclinic class of #491511
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33713	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

628 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)(9,10,16,20)(11,13,18,14)(12,23,29,39)(15,31,32,22)(17,33,34,28)(19,30,35,25)(21,36,24,40)(26,38,27,37);(1,2,3,5,4,6,7,8)(9,11,14,10,16,18,13,20)(12,24,27,34,29,21,26,17)(15,22,25,19,32,31,30,35)(23,28,38,40,39,33,37,36);(2,6)(5,8)(9,12,25,40,14,27,32,33,16,29,30,36,13,26,15,28)(10,17,31,38,18,24,35,39,20,34,22,37,11,21,19,23);(9,13,16,14)(10,18,20,11)(12,26,29,27)(15,30,32,25)(17,24,34,21)(19,31,35,22)(23,38,39,37)(28,36,33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,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,c^b=cg,d^b=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398712);