#143328

order	512
nilpotency class	4
isoclinism	isoclinic class of #143328
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=h,f^2=i,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=ehi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,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, 50524);