#490865

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	#33666	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

453 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18,24,30)(19,29,33,41)(20,34,37,40)(21,28,22,27)(23,38,32,25)(26,42,45,31)(35,48,36,46)(39,44,47,43);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,19,22,32,24,33,21,23)(18,25,28,29,30,38,27,41)(20,31,36,47,37,42,35,39)(26,40,44,46,45,34,43,48);(17,20)(18,26)(19,31)(21,35)(22,36)(23,39)(24,37)(25,40)(27,43)(28,44)(29,46)(30,45)(32,47)(33,42)(34,38)(41,48);(3,10)(6,15)(9,16)(11,14)(17,21,24,22)(18,27,30,28)(19,23,33,32)(20,35,37,36)(25,41,38,29)(26,43,45,44)(31,39,42,47)(34,46,40,48)

Polycyclic presentation:

2^9|a^2=g,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, 398061);