#500482

order	512
nilpotency class	4
isoclinism	isoclinic class of #499795
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×D₄	C₂⁴
prime radical	C₂×B(4,𝔽₂)	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

963 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)(11,15)(12,22)(13,18)(14,17)(16,23)(19,33)(20,29)(21,28)(24,36)(25,32)(26,31)(27,34)(30,37)(35,40)(38,39);(2,5)(6,7)(9,11,14,21)(10,15,17,28)(12,19,24,35)(13,25,26,20)(16,27,30,39)(18,32,31,29)(22,33,36,40)(23,34,37,38);(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,30)(18,22)(20,34)(21,35)(25,38)(26,37)(28,39)(29,33)(31,36)(32,40);(1,3,4,8)(2,6,5,7)(10,17)(15,28)(16,30)(18,31)(22,36)(27,39)(29,32)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407678);