#502323

order	512
nilpotency class	4
isoclinism	isoclinic class of #502081
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×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=ei,f^2=i,g^2=h,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=ghi,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, 409519);