#143487

order	512
nilpotency class	4
isoclinism	isoclinic class of #143450
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	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

360 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 128

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

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

Polycyclic presentation:

2^9|a^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=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eghi,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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50683);