#143810

order	512
nilpotency class	4
isoclinism	isoclinic class of #143807
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
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

449 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,d^c=dg,e^c=eh,h^c=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51006);