#490854

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	C₂³	#459
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

428 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)(19,27)(20,32)(21,36)(22,38)(23,28)(24,30)(25,29)(26,37)(31,45)(33,44)(34,40)(35,48)(39,41)(42,47)(43,46);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,19,21,30,23,31,37,22)(18,24,26,27,28,38,36,45)(20,29,33,46,34,47,35,39)(25,40,41,44,42,32,43,48);(17,20,23,34)(18,25,28,42)(19,29,31,47)(21,33,37,35)(22,39,30,46)(24,40,38,32)(26,41,36,43)(27,44,45,48);(3,10)(6,15)(9,16)(11,14)

Polycyclic presentation:

2^9|a^2=gi,b^2=e,c^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,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, 398050);