#509587

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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	#3155	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

472 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=ei,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=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, 416783);