#489064

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

716 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 156 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)(10,17)(11,22)(12,26)(13,29)(14,15)(19,33)(20,30)(21,27)(23,28)(24,31)(25,40)(32,36)(35,39)(37,38);(1,2,4,6)(3,8,7,5)(9,10)(11,18)(12,27)(13,30)(14,21)(15,20)(16,19)(17,26)(22,39)(23,38)(24,37)(25,34)(28,36)(29,33)(31,35)(32,40);(9,11,16,25)(10,18,19,34)(12,28,13,31)(14,24,15,23)(17,32,33,39)(20,38,21,37)(22,26,40,29)(27,36,30,35);(10,19)(17,33)(18,34)(20,21)(27,30)(32,39)(35,36)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396260);