#509192

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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
prime radical	#28484	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

809 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)(11,17)(12,25)(13,27)(14,29)(15,34)(16,21)(18,26)(19,28)(20,32)(22,38)(23,37)(24,39)(30,35)(31,36)(33,40);(1,2)(3,4)(5,6)(7,8)(9,11,16,15)(10,17,21,34)(12,22,13,23)(14,24,33,32)(18,35,19,36)(20,29,39,40)(25,38,27,37)(26,30,28,31);(6,8)(9,12,16,13)(10,18,21,19)(11,22,15,23)(14,30,33,31)(17,35,34,36)(20,37,39,38)(24,28,32,26)(25,40,27,29);(2,4)(9,13,16,12)(10,19,21,18)(11,23,15,22)(14,31,33,30)(17,36,34,35)(20,38,39,37)(24,26,32,28)(25,29,27,40)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^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=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416388);