#492579

order	512
nilpotency class	4
isoclinism	isoclinic class of D₈×D₁₆
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

437 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,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,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399775);