#350113

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

789 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)(3,6)(4,5)(7,8)(9,10,12,16)(11,18,19,24)(13,23,22,17)(14,21,15,20)(25,26,30,29)(27,34,31,32)(28,36,38,40)(33,39,37,35);(2,5)(6,8)(9,10)(11,14)(12,16)(13,17)(15,19)(18,20)(21,24)(22,23)(25,27,30,31)(26,32,29,34)(28,35,38,39)(33,36,37,40);(1,3,4,7)(2,6,5,8)(9,11,12,19)(10,14,16,15)(13,21,22,20)(17,24,23,18)(25,28,30,38)(26,33,29,37)(27,35,31,39)(32,36,34,40);(9,11,12,19)(10,15,16,14)(13,20,22,21)(17,24,23,18)(25,29,30,26)(27,32,31,34)(28,37,38,33)(35,36,39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,d^b=dh,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257309);