#491100

order	512
nilpotency class	4
isoclinism	isoclinic class of #491097
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	C₂²×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 456 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) = 128

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

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

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,c^b=ci,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, 398296);