#489046

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
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	C₄×C₈	C₂⁴
prime radical	#33579	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

717 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:

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

Polycyclic presentation:

2^9|b^2=ghi,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,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, 396242);