#512232

order	512
nilpotency class	4
isoclinism	isoclinic class of #512000
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=f,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gh,h^a=hi,c^b=ci,g^d=ghi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419428);