#503033

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

764 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410229);