#512438

order	512
nilpotency class	4
isoclinism	isoclinic class of #511960
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	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

530 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fhi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,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, 419634);