#502764

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

836 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,16)(13,25)(14,27)(15,18)(17,23)(19,24)(21,38)(22,34)(26,35)(28,36)(29,33)(30,31)(32,40)(37,39);(2,5)(6,7)(10,16)(11,21)(13,15)(14,28)(17,32)(19,36)(20,37)(22,39)(23,29)(24,26)(27,35)(30,40)(31,33)(34,38);(9,11,13,23,12,22,15,30)(10,17,18,20,16,31,25,34)(14,29,26,21,24,40,28,39)(19,37,35,32,27,38,36,33);(1,3,4,8)(2,6,5,7)(10,16)(17,31)(18,25)(19,27)(20,34)(32,33)(35,36)(37,38)

Polycyclic presentation:

2^9|c^2=f,d^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,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, 409960);