#512526

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1196 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1196 conjugacy classes of subgroups — too many to draw. Only its 266 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)(10,15)(12,16)(13,23)(17,25)(18,33)(19,28)(21,29)(22,35)(24,30)(26,27)(31,36)(32,40)(34,39)(37,38);(9,10,16,27,13,18,24,28,14,19,30,33,23,26,12,15)(11,17,29,38,22,32,34,36,20,31,39,40,35,37,21,25);(1,2,3,4)(9,11)(10,17)(12,21)(13,22)(14,20)(15,25)(16,29)(18,32)(19,31)(23,35)(24,34)(26,37)(27,38)(28,36)(30,39)(33,40);(5,6,7,8)(11,20)(17,31)(21,34)(22,35)(25,36)(29,39)(32,37)(38,40)

Polycyclic presentation:

2^9|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,d^c=di

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, 419722);