#146006

order	512
nilpotency class	4
isoclinism	isoclinic class of #143943
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#459
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

288 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=ei,c^2=d,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fh,h^b=hi,e^c=eg,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53202);