#499064

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1276 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)(3,4)(6,7)(9,10,14,18)(11,19,21,37)(12,23,27,40)(13,28,26,17)(15,30,31,20)(16,33,35,24)(22,38,29,34)(25,36,39,32);(5,6)(7,8)(11,20)(12,24)(13,26)(15,19)(16,23)(17,28)(21,30)(22,39)(25,29)(27,33)(31,37)(32,38)(34,36)(35,40);(2,4)(5,7,8,6)(9,11)(10,15)(12,25)(13,29)(14,21)(16,34)(17,36)(18,31)(19,23)(20,24)(22,26)(27,39)(28,32)(30,33)(35,38)(37,40);(11,21)(15,31)(19,37)(20,30)(22,29)(25,39)(32,36)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406260);