#490474

order	512
nilpotency class	4
isoclinism	isoclinic class of #490474
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

576 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 150 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)(4,7)(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);(2,5)(3,8)(6,7)(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);(11,20)(17,31)(21,34)(22,35)(25,36)(29,39)(32,37)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397670);