#490435

order	512
nilpotency class	4
isoclinism	isoclinic class of #490434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

495 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

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, 397631);