#509196

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

Polycyclic presentation:

2^9|a^2=e,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, 416392);