#490824

order	512
nilpotency class	4
isoclinism	isoclinic class of #490801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

467 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)(9,10,15,21)(11,20,24,32)(12,26,28,39)(13,22,16,19)(14,17,25,31)(18,33,35,23)(27,40,30,36)(29,37,38,34);(1,2,3,5,4,6,7,8)(9,11,16,25,15,24,13,14)(10,17,22,32,21,31,19,20)(12,23,30,38,28,33,27,29)(18,26,37,40,35,39,34,36);(9,12,15,28)(10,18,21,35)(11,23,24,33)(13,27,16,30)(14,29,25,38)(17,26,31,39)(19,34,22,37)(20,36,32,40);(2,6)(5,8)(9,13,15,16)(10,19,21,22)(11,14,24,25)(12,27,28,30)(17,20,31,32)(18,34,35,37)(23,29,33,38)(26,36,39,40)

Polycyclic presentation:

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

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, 398020);