#143188

order	512
nilpotency class	4
isoclinism	isoclinic class of #143188
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
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

395 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18)(19,30)(20,35)(21,25)(22,27)(23,29)(24,32)(26,36)(28,37)(31,43)(33,46)(34,40)(38,44)(39,45)(41,48)(42,47);(6,15)(9,16)(17,19)(18,24)(20,31)(21,33)(22,39)(23,34)(25,43)(26,45)(27,41)(28,46)(29,42)(30,35)(32,36)(37,47)(38,48)(40,44);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(18,25)(19,31)(20,36)(21,22)(24,40)(26,28)(27,29)(30,42)(32,33)(34,48)(37,38)(39,47)(41,45)(43,46)

Polycyclic presentation:

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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50384);