#502828

order	512
nilpotency class	4
isoclinism	isoclinic class of #501096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#3496	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

969 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)(3,7)(4,5)(6,8)(9,10,14,18)(11,15,20,29)(12,16,24,33)(13,17,27,36)(19,37,38,28)(21,30,25,34)(22,31,39,40)(23,35,26,32);(2,5)(6,7)(9,11,14,20)(10,15,18,29)(12,22,24,39)(13,25,27,21)(16,31,33,40)(17,34,36,30)(19,26,38,23)(28,35,37,32);(9,12)(10,16)(11,19,20,38)(13,26,27,23)(14,24)(15,28,29,37)(17,35,36,32)(18,33)(21,39)(22,25)(30,40)(31,34);(1,3,4,8)(2,6,5,7)(11,20)(15,29)(19,38)(21,25)(22,39)(28,37)(30,34)(31,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,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, 410024);