#509586

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	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

464 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)(9,10,14,20)(11,15,26,17)(12,28,19,24)(13,25,18,31)(16,22,34,23)(21,40,27,37)(29,30,36,38)(32,35,33,39);(1,3,5,7)(2,6,8,4)(9,11)(10,17)(12,29)(13,31)(14,26)(15,20)(16,21)(18,25)(19,36)(22,37)(23,40)(24,30)(27,34)(28,38)(32,35)(33,39);(1,4,5,6)(2,7,8,3)(9,12)(10,18)(11,24)(13,20)(14,19)(15,35)(16,38)(17,39)(21,29)(22,32)(23,33)(25,37)(26,28)(27,36)(30,34)(31,40);(2,8)(3,7)(9,13)(10,19)(11,25)(12,20)(14,18)(15,36)(16,32)(17,29)(21,39)(22,38)(23,30)(24,37)(26,31)(27,35)(28,40)(33,34)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,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, 416782);