#499259

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
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	C₂²×C₈	C₂⁴
prime radical	#60695	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

830 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406455);