#143806

order	512
nilpotency class	4
isoclinism	isoclinic class of #143803
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	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 428 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51002);