#499473

order	512
nilpotency class	4
isoclinism	isoclinic class of #498860
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

603 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)(3,6)(4,5)(7,8)(9,10,15,17,14,18,12,19)(11,16,24,33,25,30,22,31)(13,20,28,34,27,35,26,36)(21,37,38,29,39,40,23,32);(2,5)(6,8)(9,11)(10,16)(12,22)(13,21)(14,25)(15,24)(17,33)(18,30)(19,31)(20,37)(23,26)(27,39)(28,38)(29,34)(32,36)(35,40);(10,17)(11,21)(12,15)(13,26)(16,29)(18,19)(22,38)(23,24)(25,39)(27,28)(30,32)(31,40)(33,37)(34,36);(1,3,4,7)(2,6,5,8)(10,18)(16,30)(17,19)(20,35)(29,32)(31,33)(34,36)(37,40)

Polycyclic presentation:

2^9|a^2=fi,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,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, 406669);