#490551

order	512
nilpotency class	4
isoclinism	isoclinic class of #490479
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 397747);