#501497

order	512
nilpotency class	4
isoclinism	isoclinic class of #501014
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

604 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)(9,10,14,19,16,20,13,15)(11,17,23,32,24,33,22,30)(12,25,27,34,28,35,29,18)(21,26,36,39,38,40,37,31);(2,5)(6,7)(9,11)(10,17)(12,26)(13,22)(14,23)(15,30)(16,24)(18,21)(19,32)(20,33)(25,36)(27,39)(28,40)(29,31)(34,38)(35,37);(9,12,16,28)(10,18,20,34)(11,21,24,38)(13,27,14,29)(15,25,19,35)(17,31,33,39)(22,36,23,37)(26,32,40,30);(1,3,4,8)(2,6,5,7)(9,13,16,14)(10,15,20,19)(11,22,24,23)(12,27,28,29)(17,30,33,32)(18,25,34,35)(21,36,38,37)(26,39,40,31)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408693);