#503063

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60718	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

749 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410259);