#504165

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
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×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

526 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=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, 411361);