#145935

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:

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

Polycyclic presentation:

2^9|a^2=de,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=egi,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, 53131);