#509900

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
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	#3061	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 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)(3,4)(5,7)(6,8)(9,10,17,24,33,29,35,36,14,12,15,20,16,21,37,34)(11,22,28,13,26,38,40,31,25,18,19,30,27,23,39,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,18)(12,22)(13,20)(14,25)(15,19)(16,27)(17,28)(21,38)(23,29)(24,30)(26,33)(31,34)(32,36)(35,40)(37,39);(1,4,5,6)(2,3,7,8)(9,12,14,10)(11,23,25,38)(13,28,30,19)(15,34,17,36)(16,21,33,29)(18,27,22,26)(20,35,24,37)(31,39,32,40);(2,7)(3,8)(9,13,14,30)(10,19,12,28)(11,24,25,20)(15,18,17,22)(16,31,33,32)(21,39,29,40)(23,37,38,35)(26,34,27,36)

Polycyclic presentation:

2^9|a^2=fi,c^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417096);