#490443

order	512
nilpotency class	4
isoclinism	isoclinic class of #490434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

486 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397639);