#483584

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3169	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

506 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cgi,d^a=di,e^a=eg,d^b=di,e^b=eg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390780);