#500710

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3285	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1084 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=hi

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, 407906);