#504257

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
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	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=f,d^2=ei,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,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, 411453);