#512524

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1112 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 282 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=g,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419720);