#145044

order	512
nilpotency class	4
isoclinism	isoclinic class of #145023
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=d,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,f^b=fh,e^c=egi,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52240);