#489061

order	512
nilpotency class	4
isoclinism	isoclinic class of #488912
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	C₄×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

708 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)(9,10,15,21)(11,22,27,35)(12,20,14,18)(13,29,33,37)(16,25,17,24)(19,28,39,32)(23,40,26,38)(30,36,31,34);(1,2,4,6)(3,8,7,5)(9,11)(10,16)(12,23)(13,30)(14,26)(15,27)(17,21)(18,34)(19,38)(20,36)(22,29)(24,32)(25,28)(31,33)(35,37)(39,40);(9,11)(10,17)(12,23)(13,31)(14,26)(15,27)(16,21)(18,36)(19,38)(20,34)(22,29)(24,28)(25,32)(30,33)(35,37)(39,40);(9,12,15,14)(10,18,21,20)(11,23,27,26)(13,28,33,32)(16,34,17,36)(19,37,39,29)(22,38,35,40)(24,30,25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396257);