#492589

order	512
nilpotency class	4
isoclinism	isoclinic class of #492504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×D₄	C₂×D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

434 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:

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

Polycyclic presentation:

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

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, 399785);