#512213

order	512
nilpotency class	4
isoclinism	isoclinic class of #511960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

590 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)(3,6)(4,5)(7,8)(9,10,18,11)(12,27,22,14)(13,29,33,38)(15,26,19,21)(16,24,17,23)(20,34,25,32)(28,31,37,35)(30,39,40,36);(2,5)(6,8)(9,11,15,12,17,23,14,26,18,10,19,22,16,24,27,21)(13,25,31,28,32,38,30,39,33,20,35,37,34,29,40,36);(1,3,4,7)(2,6,5,8)(9,12,14,10,16,21,15,23,18,22,27,11,17,26,19,24)(13,28,30,20,34,36,31,38,33,37,40,25,32,39,35,29);(9,13)(10,20)(11,25)(12,28)(14,30)(15,31)(16,34)(17,32)(18,33)(19,35)(21,36)(22,37)(23,38)(24,29)(26,39)(27,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=e,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419409);