#143493

order	512
nilpotency class	4
isoclinism	isoclinic class of #143438
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#459
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

354 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=gi,e^2=gh,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fg,g^c=gi,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, 50689);