#510053

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

494 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=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, 417249);