#507423

order	512
nilpotency class	4
isoclinism	isoclinic class of #501764
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

548 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fh,g^c=gi,h^c=hi

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, 414619);