#500530

order	512
nilpotency class	4
isoclinism	isoclinic class of #499803
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

959 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)(9,10,13,17)(11,19,21,15)(12,23,24,39)(14,18,26,32)(16,30,31,25)(20,35,36,38)(22,33,27,29)(28,37,34,40);(2,5)(6,7)(9,11,13,21)(10,15,17,19)(12,20,24,36)(14,27,26,22)(16,28,31,34)(18,33,32,29)(23,38,39,35)(25,37,30,40);(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,31)(18,23)(19,34)(21,36)(22,37)(26,30)(27,40)(29,38)(32,39)(33,35);(1,3,4,8)(2,6,5,7)(10,17)(11,21)(16,31)(18,32)(20,36)(22,27)(23,39)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=ei,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,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, 407726);