#491062

order	512
nilpotency class	4
isoclinism	isoclinic class of #491017
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	#466	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

422 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)(17,18,23,28)(19,29,32,46)(20,27,22,25)(21,34,37,42)(24,39,41,31)(26,33,44,36)(30,47,38,43)(35,45,48,40);(1,3,5,8,6,13,15,16)(2,7,9,11,10,4,12,14);(1,4)(2,8)(3,12)(5,14)(6,7)(9,13)(10,16)(11,15)(17,19)(18,24)(20,30)(21,35)(22,38)(23,32)(25,40)(26,43)(27,45)(28,41)(29,34)(31,36)(33,39)(37,48)(42,46)(44,47);(3,13)(4,7)(8,16)(11,14)(17,20,23,22)(18,25,28,27)(19,30,32,38)(21,33,37,36)(24,40,41,45)(26,42,44,34)(29,43,46,47)(31,35,39,48)

Polycyclic presentation:

2^9|a^2=gi,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=dg,d^c=dg,f^c=fh,h^c=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398258);