#146097

order	512
nilpotency class	4
isoclinism	isoclinic class of #146090
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2071
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#486
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

225 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53293);