#218316

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

498 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^b=dh,d^c=di,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125512);