#502374

order	512
nilpotency class	4
isoclinism	isoclinic class of #502081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=i,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=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, 409570);