#126966

order	512
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#95
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,e^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fi,d^b=di,e^b=eg

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, 34162);