#145875

order	512
nilpotency class	4
isoclinism	isoclinic class of #145862
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2058
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#450
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

285 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 86 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)(9,10,12,16)(11,21)(13,19,27,33)(14,18,28,32)(15,29,23,20)(17,34)(22,37)(24,31)(25,35)(26,39)(30,36)(38,40);(10,16)(11,22,26,40)(13,14)(15,23)(17,35,31,36)(18,33)(19,32)(21,37,39,38)(24,30,34,25)(27,28);(2,5)(3,8)(6,7)(9,11,23,25,12,26,15,30)(10,17,29,37,16,31,20,38)(13,22,28,34,27,40,14,24)(18,36,33,39,32,35,19,21)

Polycyclic presentation:

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

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, 53071);