#490528

order	512
nilpotency class	4
isoclinism	isoclinic class of #490434
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 ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

458 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,29)(20,26)(21,27)(22,36)(23,39)(24,31)(25,32)(28,38)(30,43)(33,41)(34,46)(35,40)(37,44)(42,48)(45,47);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19)(18,24)(20,33)(21,31)(22,32)(23,30)(25,40)(26,43)(27,29)(28,42)(34,48)(35,47)(36,45)(37,38)(39,41)(44,46);(17,19)(18,25)(20,30)(21,31)(22,37)(23,33)(24,40)(26,44)(27,45)(28,41)(29,36)(32,38)(34,47)(35,48)(39,42)(43,46);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,20,21,23)(18,26,27,39)(19,30,31,33)(22,34,35,38)(24,41,29,43)(25,44,45,42)(28,36,46,40)(32,37,47,48)

Polycyclic presentation:

2^9|b^2=ei,d^2=gi,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=di,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397724);