#501320

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×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

684 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)(4,6)(5,8)(9,10,14,19,15,17,29,34,13,18,27,35,12,20,26,28)(11,21,25,36,30,33,40,39,24,16,31,32,23,22,37,38);(1,3,4,8)(2,5,6,7)(9,11,13,24)(10,16,18,21)(12,23,15,30)(14,25,27,31)(17,22,20,33)(19,32,35,36)(26,37,29,40)(28,39,34,38);(9,10)(11,22)(12,17)(13,18)(14,28)(15,20)(16,30)(19,26)(21,23)(24,33)(25,32)(27,34)(29,35)(31,36)(37,38)(39,40);(2,6)(5,7)(9,12,13,15)(10,17,18,20)(11,23,24,30)(14,26,27,29)(16,22,21,33)(19,34,35,28)(25,37,31,40)(32,38,36,39)

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=i,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, 408516);