#491094

order	512
nilpotency class	4
isoclinism	isoclinic class of #491091
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

443 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 130 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)(9,10,14,19)(11,20,24,36)(12,17,15,26)(13,28,29,40)(16,30,32,23)(18,27,33,21)(22,34,25,39)(31,37,35,38);(9,11,21,37,15,25,29,30,14,24,27,38,12,22,13,23)(10,16,28,34,17,31,33,36,19,32,40,39,26,35,18,20);(2,5)(3,8)(6,7)(9,10)(11,16)(12,26)(13,18)(14,19)(15,17)(20,23)(21,28)(22,35)(24,32)(25,31)(27,40)(29,33)(30,36)(34,37)(38,39);(2,6)(5,7)(9,12,14,15)(10,17,19,26)(11,22,24,25)(13,27,29,21)(16,31,32,35)(18,28,33,40)(20,34,36,39)(23,38,30,37)

Polycyclic presentation:

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

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, 398290);