#143928

order	512
nilpotency class	4
isoclinism	isoclinic class of #143907
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 341 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=g,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,e^c=eh,f^c=fg,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, 51124);