#145973

order	512
nilpotency class	4
isoclinism	isoclinic class of #145934
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2067
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#468
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

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

normal subgroups only

This group has 255 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:

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

Polycyclic presentation:

2^9|a^2=deh,b^2=dg,c^2=i,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,f^b=fi,d^c=dg,e^c=eh,f^c=fg,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, 53169);