#512415

order	512
nilpotency class	4
isoclinism	isoclinic class of #511887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

568 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=f,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gh,h^a=hi,g^d=gh,h^d=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, 419611);