#502122

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	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

787 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)(9,10,16,22,29,21,30,36,14,11,15,19,13,18,33,38)(12,23,40,35,25,39,31,17,26,24,27,34,28,37,32,20);(2,4)(5,6)(7,8)(9,11,15,22,29,18,33,36,14,10,16,19,13,21,30,38)(12,24,27,35,25,37,32,17,26,23,40,34,28,39,31,20);(6,8)(9,12)(10,17)(11,20)(13,25)(14,26)(15,31)(16,32)(18,34)(19,37)(21,35)(22,39)(23,36)(24,38)(27,30)(28,29)(33,40);(2,4)(9,13,14,29)(10,18,11,21)(12,25,26,28)(15,30,16,33)(17,34,20,35)(19,36,22,38)(23,39,24,37)(27,32,40,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gh,h^c=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, 409318);