#507473

order	512
nilpotency class	4
isoclinism	isoclinic class of #501766
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

498 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=g,c^2=f,d^2=ei,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414669);