#499414

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	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 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)(3,8)(4,6)(5,7)(9,10,12,19,15,18,14,17)(11,16,22,31,24,30,28,32)(13,20,25,35,27,34,26,33)(21,36,37,39,38,40,23,29);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,22)(13,21)(14,28)(15,24)(17,32)(18,30)(19,31)(20,36)(23,26)(25,37)(27,38)(29,33)(34,40)(35,39);(1,4,5,3)(2,6,7,8)(10,17)(11,21)(12,14)(13,26)(16,29)(18,19)(20,34)(22,23)(24,38)(25,27)(28,37)(30,39)(31,40)(32,36);(10,18)(16,30)(17,19)(20,34)(29,39)(31,32)(33,35)(36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^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, 406610);