#512525

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1138 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1138 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)(9,10)(11,19)(12,24)(13,16)(14,27)(15,22)(17,20)(18,25)(21,31)(23,38)(26,39)(28,36)(29,40)(30,35)(32,34)(33,37);(9,11,20,34,14,23,26,36,13,21,35,40,25,37,12,22)(10,15,24,33,18,29,30,31,16,28,39,38,27,32,17,19);(1,2,3,4)(11,21)(15,28)(19,31)(22,36)(23,37)(29,32)(33,38)(34,40);(5,6,7,8)(10,16)(15,28)(17,30)(18,27)(19,31)(24,39)(29,32)(33,38)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=gi,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, 419721);