#465358

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

943 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=eg,c^2=efi,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372554);