#359266

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

982 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)(3,6)(4,5)(7,8)(9,10,15,20,16,22,14,21)(11,23,27,36,28,40,26,17)(12,29,30,38,31,35,25,18)(13,19,24,39,33,37,32,34);(2,5)(6,8)(9,11,14,26,16,28,15,27)(10,17,20,23,22,36,21,40)(12,13,25,32,31,33,30,24)(18,37,29,34,38,19,35,39);(1,3,4,7)(2,6,5,8)(9,12,14,25,16,31,15,30)(10,18,20,29,22,38,21,35)(11,24,26,13,28,32,27,33)(17,34,23,19,36,39,40,37);(9,13,15,24,16,33,14,32)(10,19,20,39,22,37,21,34)(11,25,27,12,28,30,26,31)(17,35,23,18,36,29,40,38)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266462);