#489051

order	512
nilpotency class	4
isoclinism	isoclinic class of #488895
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	#33582	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

742 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:

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

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,d^a=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396247);