#484088

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#508	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

453 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=dhi,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391284);