#490867

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

442 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)(17,18,23,29)(19,28,32,40)(20,34,36,47)(21,30,24,27)(22,25,33,39)(26,41,43,31)(35,48,38,44)(37,45,46,42);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,19,24,33,23,32,21,22)(18,25,30,40,29,39,27,28)(20,31,38,46,36,41,35,37)(26,34,45,48,43,47,42,44);(17,20)(18,26)(19,31)(21,35)(22,37)(23,36)(24,38)(25,34)(27,42)(28,44)(29,43)(30,45)(32,41)(33,46)(39,47)(40,48);(3,10)(6,15)(9,16)(11,14)(17,21,23,24)(18,27,29,30)(19,22,32,33)(20,35,36,38)(25,28,39,40)(26,42,43,45)(31,37,41,46)(34,44,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,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, 398063);