#502500

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	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

720 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efh,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,d^c=di

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, 409696);