#145046

order	512
nilpotency class	4
isoclinism	isoclinic class of #145024
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	C₂³	#450
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

321 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 76 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)(9,10,15,21)(11,22,26,34)(12,27,32,39)(13,20,14,18)(16,25,19,24)(17,28,37,30)(23,35,40,36)(29,38,31,33);(1,2)(3,5)(4,6)(7,8)(9,11,14,25,15,26,13,24)(10,16,20,34,21,19,18,22)(12,28,29,23,32,30,31,40)(17,27,35,33,37,39,36,38);(2,5)(3,8)(6,7)(9,12,13,29,15,32,14,31)(10,17,18,35,21,37,20,36)(11,23,24,30,26,40,25,28)(16,33,22,39,19,38,34,27)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=d,d^2=i,e^2=gi,f^2=h,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=fhi,e^c=egi,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52242);