#490817

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	#33582	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=e,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, 398013);