#500585

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

849 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)(11,15)(12,16)(13,28)(14,19)(17,25)(18,26)(20,30)(21,38)(22,31)(23,32)(24,33)(27,35)(29,37)(34,40)(36,39);(2,5)(6,7)(9,11)(10,15)(12,24)(13,20)(14,23)(16,33)(17,36)(18,31)(19,32)(21,27)(22,26)(25,39)(28,30)(29,34)(35,38)(37,40);(10,16)(11,20)(12,25)(13,26)(15,29)(17,18)(19,35)(21,24)(22,23)(27,40)(28,34)(30,38)(31,33)(32,36);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|d^2=e,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, 407781);