#490579

order	512
nilpotency class	4
isoclinism	isoclinic class of #490474
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

470 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eghi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^c=di,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397775);