#483631

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	#3106	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

462 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)(3,5)(4,6)(7,8)(9,10,13,18,14,12,24,26,15,19,28,35,29,27,17,33)(11,16,22,32,23,25,30,39,21,31,37,40,38,20,34,36);(2,5)(3,8)(6,7)(9,11)(10,16)(12,25)(13,22)(14,23)(15,21)(17,34)(18,32)(19,31)(20,27)(24,30)(26,39)(28,37)(29,38)(33,36)(35,40);(9,12,13,26,14,19,24,35,15,27,28,33,29,10,17,18)(11,20,22,36,23,16,30,32,21,25,37,39,38,31,34,40);(2,6)(5,7)(11,21)(16,31)(20,25)(22,37)(23,38)(30,34)(32,40)(36,39)

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=dg,e^a=eg,c^b=ci,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, 390827);