#491564

order	512
nilpotency class	4
isoclinism	isoclinic class of #491556
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

600 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)(11,20)(12,17)(13,26)(14,29)(15,19)(16,23)(18,27)(21,33)(22,34)(24,40)(25,36)(28,30)(31,37)(32,39)(35,38);(1,2,3,5,4,6,7,8);(9,11,21,37,14,24,13,23,15,25,27,38,12,22,28,39)(10,16,30,40,17,31,18,20,19,32,26,34,29,35,33,36);(2,6)(5,8)(9,12,15,14)(10,17,19,29)(11,22,25,24)(13,21,28,27)(16,31,32,35)(18,26,33,30)(20,34,36,40)(23,37,39,38)

Polycyclic presentation:

2^9|b^2=e,c^2=fhi,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=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398760);