#415329

order	512
nilpotency class	3
isoclinism	isoclinic class of #397950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1296 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:

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

Polycyclic presentation:

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

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, 322525);