#509197

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=dei,f^c=fh,h^c=hi,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, 416393);