#500459

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61287	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

779 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,26)(13,18)(14,17)(15,19)(16,22)(21,36)(23,34)(24,32)(25,33)(27,37)(28,30)(29,38)(31,35)(39,40);(2,5)(6,7)(9,10)(11,21)(12,17)(13,28)(14,26)(15,19)(16,23)(18,30)(20,36)(22,34)(24,40)(25,38)(27,35)(29,33)(31,37)(32,39);(9,11,15,25)(10,16,19,32)(12,24,14,22)(13,29,31,36)(17,33,26,20)(18,34,35,40)(21,37,38,28)(23,27,39,30);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|c^2=i,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=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, 407655);