#492510

order	512
nilpotency class	4
isoclinism	isoclinic class of #492504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19973	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

496 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)(3,5)(4,7)(6,8)(10,15)(11,19)(12,23)(13,26)(16,33)(17,37)(18,32)(20,36)(21,34)(22,29)(24,28)(27,30)(31,35)(38,39);(9,10,14,18)(11,20,22,33)(12,24,27,37)(13,28,26,17)(15,30,32,23)(16,34,36,21)(19,39,29,31)(25,38,40,35);(2,5)(3,8)(6,7);(9,11,14,22)(10,16,18,36)(12,25,27,40)(13,29,26,19)(15,31,32,39)(17,38,28,35)(20,37,33,24)(21,30,34,23)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399706);