#500545

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×H(5,𝔽₂)	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

927 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)(11,15,20,30)(12,22,24,34)(13,25,28,17)(16,27,32,23)(19,35,36,40)(21,33,26,31)(29,38,39,37);(2,5)(6,7)(9,11,14,20)(10,15,18,30)(12,19,24,36)(13,26,28,21)(16,29,32,39)(17,33,25,31)(22,35,34,40)(23,37,27,38);(9,12)(10,16)(11,19)(13,27)(14,24)(15,29)(17,34)(18,32)(20,36)(21,38)(22,25)(23,28)(26,37)(30,39)(31,40)(33,35);(1,3,4,8)(2,6,5,7)(11,20)(15,30)(19,36)(21,26)(29,39)(31,33)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=e,f^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,f^c=fi,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, 407741);