#501316

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2184	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

743 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)(4,6)(5,8)(9,10,13,18,26,36,39,34,12,17,25,35,14,19,27,16)(11,20,23,29,24,32,38,33,22,15,28,30,21,31,37,40);(1,3,4,8)(2,5,6,7)(9,11,12,22)(10,15,17,20)(13,23,25,28)(14,21,26,24)(16,33,34,40)(18,30,35,29)(19,32,36,31)(27,37,39,38);(10,16)(11,21)(13,27)(14,26)(15,29)(17,34)(18,19)(20,30)(22,24)(23,28)(25,39)(31,40)(32,33)(35,36);(2,6)(5,7)(10,17)(15,20)(16,34)(18,35)(19,36)(29,30)(31,32)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408512);