#489043

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
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

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

normal subgroups only

This group has 717 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)(10,15)(11,19)(12,22)(13,25)(17,31)(18,35)(20,38)(21,37)(23,27)(24,28)(26,29)(30,33)(34,36)(39,40);(1,2,4,6)(3,8,7,5)(9,10,14,17)(11,16,21,32)(12,23,26,35)(13,27,25,18)(15,29,31,22)(19,30,37,39)(20,36,28,34)(24,33,38,40);(9,11,14,21)(10,16,17,32)(12,24,26,38)(13,28,25,20)(15,30,31,39)(18,36,27,34)(19,29,37,22)(23,33,35,40);(10,17)(15,31)(16,32)(18,27)(23,35)(30,39)(33,40)(34,36)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,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=fh,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, 396239);