#512439

order	512
nilpotency class	4
isoclinism	isoclinic class of #511963
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

518 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)(12,18)(13,14)(17,28)(19,34)(20,31)(21,30)(23,37)(24,27)(25,26)(29,35)(32,40)(36,39);(1,3,4,8)(2,5,6,7)(10,17)(16,28)(19,32)(20,21)(29,39)(30,31)(34,40)(35,36);(9,10,18,31,14,21,26,28,15,17,25,30,13,20,12,16)(11,19,33,34,22,35,37,39,24,32,38,40,27,36,23,29);(9,11)(10,19)(12,23)(13,27)(14,22)(15,24)(16,29)(17,32)(18,33)(20,36)(21,35)(25,38)(26,37)(28,39)(30,40)(31,34)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,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,c^b=ci,g^d=gi

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, 419635);