#143327

order	512
nilpotency class	4
isoclinism	isoclinic class of #143327
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10056
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

362 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(3,11,14,28)(4,16)(5,20)(7,22,12,30)(8,17)(10,21)(13,29,15,18)(19,27)(23,32,24,25)(26,31);(1,2)(3,12,14,7)(4,17)(5,21)(6,9)(8,16)(10,20)(11,25,28,32)(13,23,15,24)(18,22,29,30)(19,31)(26,27);(1,3)(2,7)(4,18)(5,13)(6,14)(8,25)(9,12)(10,23)(11,27)(15,20)(16,28)(17,30)(19,29)(21,24)(22,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,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, 50523);