#483523

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

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	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

468 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,c^a=ci,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, 390719);