#502131

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	#24778	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

792 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)(9,10,14,19,27,32,29,28,12,17,13,18,15,20,31,30)(11,21,25,34,40,36,33,22,23,16,24,35,26,38,37,39);(2,4)(5,6)(7,8)(9,11,13,24,27,40,31,37,12,23,14,25,15,26,29,33)(10,16,18,34,32,38,30,22,17,21,19,35,20,36,28,39);(6,8)(9,10,12,17)(11,22,23,39)(13,28,14,30)(15,32,27,20)(16,33,21,37)(18,31,19,29)(24,38,25,36)(26,35,40,34);(2,4)(11,23)(16,21)(22,39)(24,25)(26,40)(33,37)(34,35)(36,38)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,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, 409327);