#491076

order	512
nilpotency class	4
isoclinism	isoclinic class of #491025
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	#33817	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

430 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ei,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=dgi,d^c=dg,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, 398272);