#129095

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	D₈
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#107
prime radical	C₂³	#427
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,d^b=dg,e^b=eg,g^b=gi

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36291);