#500589

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
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	#13319	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

801 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)(12,17)(13,16)(14,29)(15,22)(18,28)(19,26)(21,32)(23,33)(24,31)(25,34)(27,40)(30,38)(35,39)(36,37);(2,5)(6,7)(9,11)(10,15)(12,22)(13,27)(14,21)(16,33)(17,20)(18,36)(19,32)(23,25)(24,26)(28,38)(29,31)(30,35)(34,40)(37,39);(10,16)(11,21)(13,28)(14,26)(15,30)(17,34)(18,19)(20,36)(22,24)(23,27)(25,39)(29,35)(31,40)(32,33);(1,3,4,8)(2,6,5,7)(10,17)(15,20)(16,34)(18,35)(19,29)(30,36)(31,32)(33,40)

Polycyclic presentation:

2^9|d^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fh,g^c=gi,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, 407785);