#489372

order	512
nilpotency class	4
isoclinism	isoclinic class of #489365
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

774 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)(12,17)(13,26)(14,30)(15,20)(16,23)(18,25)(19,28)(22,37)(24,33)(27,32)(29,36)(31,38)(34,40)(35,39);(3,7)(5,8)(9,11,15,24)(10,16,20,35)(12,22,14,31)(13,27,29,40)(17,32,30,34)(18,37,19,38)(21,36,33,26)(23,25,39,28);(1,2)(3,8)(4,6)(5,7)(9,12,15,14)(10,17,20,30)(11,22,24,31)(13,25,29,28)(16,32,35,34)(18,36,19,26)(21,37,33,38)(23,27,39,40);(9,13,12,25,15,29,14,28)(10,18,30,26,20,19,17,36)(11,23,31,40,24,39,22,27)(16,33,32,38,35,21,34,37)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396568);