#489368

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
prime radical	#62420	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

775 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)(9,10,16,21)(11,22,26,40)(12,18,15,30)(13,20,27,37)(14,19,28,38)(17,32,36,25)(23,34,31,39)(24,35,29,33);(3,7)(5,8)(9,11)(10,17)(12,23)(13,25)(14,29)(15,31)(16,26)(18,33)(19,22)(20,39)(21,36)(24,28)(27,32)(30,35)(34,37)(38,40);(1,2)(3,8)(4,6)(5,7)(9,12,16,15)(10,18,21,30)(11,23,26,31)(13,14,27,28)(17,33,36,35)(19,37,38,20)(22,34,40,39)(24,25,29,32);(9,13,15,28,16,27,12,14)(10,19,18,37,21,38,30,20)(11,24,23,25,26,29,31,32)(17,34,35,22,36,39,33,40)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=g,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, 396564);