#488954

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	#61091	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

896 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:

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

Polycyclic presentation:

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

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, 396150);