#490708

order	512
nilpotency class	4
isoclinism	isoclinic class of #490699
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

470 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 140 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)(12,23,27,38)(14,30,32,22)(16,33,34,26)(19,36,35,29)(21,31,24,40)(25,37,28,39);(1,2,3,5,4,6,7,8)(9,11,13,10,15,18,20,17)(12,24,25,34,27,21,28,16)(14,22,29,19,32,30,36,35)(23,26,39,40,38,33,37,31);(2,6)(5,8)(9,12,15,27)(10,16,17,34)(11,21,18,24)(13,25,20,28)(14,31,32,40)(19,37,35,39)(22,38,30,23)(26,36,33,29);(10,17)(11,18)(16,34)(19,35)(21,24)(22,30)(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,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397904);