#145146

order	512
nilpotency class	4
isoclinism	isoclinic class of #145145
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

353 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=dg,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=cf,e^b=ei,f^b=fh,d^c=di,e^c=egi,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52342);