#504168

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

530 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)(9,10,14,21)(11,19,26,18)(12,27,28,37)(13,17,31,35)(15,22,16,33)(20,24,36,30)(23,39,32,38)(25,40,29,34);(1,3,5,4)(2,6,7,8)(10,17)(11,23)(13,30)(15,16)(18,37)(19,27)(20,22)(21,35)(24,31)(25,29)(26,32)(33,36)(34,38)(39,40);(1,4,5,3)(2,6,7,8)(9,11,14,26)(10,18,21,19)(12,24,28,30)(13,25,31,29)(15,32,16,23)(17,34,35,40)(20,37,36,27)(22,39,33,38);(9,12,14,28)(10,19,21,18)(11,24,26,30)(13,23,31,32)(15,29,16,25)(17,27,35,37)(20,40,36,34)(22,39,33,38)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,d^2=i,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, 411364);