#501558

order	512
nilpotency class	4
isoclinism	isoclinic class of #501014
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×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

535 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:

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

Polycyclic presentation:

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

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, 408754);