#509582

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ei,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, 416778);