#504790

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×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

563 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)(9,10,16,33,31,17,25,18,15,22,13,20,14,21,11,23)(12,26,24,35,28,40,32,19,30,39,27,36,29,38,34,37);(2,5)(6,7)(9,11)(10,17)(12,24)(13,31)(14,16)(15,25)(18,23)(19,35)(21,22)(26,39)(27,30)(28,34)(29,32)(36,37);(10,18)(11,16)(13,25)(14,31)(17,33)(19,26)(20,21)(22,23)(24,34)(27,32)(28,29)(35,40)(36,38)(37,39);(1,3,4,8)(2,6,5,7)(9,12)(10,19)(11,24)(13,32)(14,28)(15,30)(16,34)(17,35)(18,26)(20,38)(21,36)(22,37)(23,39)(25,27)(29,31)(33,40)

Polycyclic presentation:

2^9|a^2=ei,d^2=efh,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi,e^d=eh,f^d=fh,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, 411986);