#145483

order	512
nilpotency class	4
isoclinism	isoclinic class of #145482
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
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 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

293 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 62 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)(9,10,16,22)(11,23,28,18)(12,29,30,36)(13,20,31,38)(14,19,32,40)(15,33,24,21)(17,26,37,25)(27,39,34,35);(3,8)(5,6)(9,11)(10,17)(12,13)(14,25)(15,34)(16,28)(18,38)(19,35)(20,23)(21,29)(22,37)(24,27)(26,32)(30,31)(33,36)(39,40);(2,5)(3,8)(6,7)(9,12,13,11,24,25,32,34,16,30,31,28,15,26,14,27)(10,18,19,36,33,35,38,17,22,23,40,29,21,39,20,37)

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=gi,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fgi,g^b=gi,h^b=hi,e^c=egh

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, 52679);