#491835

order	512
nilpotency class	4
isoclinism	isoclinic class of #491760
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3283	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

439 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)(3,8)(6,7)(9,10,13,16)(11,20,22,37)(12,18,15,30)(14,27,26,38)(17,33,31,23)(19,25,32,29)(21,36,24,40)(28,35,39,34);(1,2,4,7)(3,6,8,5)(10,16)(17,31)(18,30)(19,32)(20,37)(27,38)(34,35)(36,40);(9,11,13,22)(10,17,16,31)(12,24,15,21)(14,28,26,39)(18,35,30,34)(19,36,32,40)(20,38,37,27)(23,25,33,29);(2,6)(3,8)(5,7)(9,12,13,15)(10,18,16,30)(11,21,22,24)(14,25,26,29)(17,34,31,35)(19,27,32,38)(20,36,37,40)(23,39,33,28)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399031);