#504259

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

513 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411455);