#146051

order	512
nilpotency class	4
isoclinism	isoclinic class of #143935
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 88 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)(10,13)(14,16)(17,18,20,24,27,31,32,29,19,22,26,30,21,25,28,23);(2,5)(6,8)(10,14)(12,15)(13,16)(18,22)(23,29)(24,30)(25,31);(1,3,4,7)(2,6,5,8)(9,10)(11,13)(12,16)(14,15)(18,23)(20,28)(21,27)(22,29)(24,25)(26,32)(30,31)

Polycyclic presentation:

2^9|a^2=eh,c^2=di,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fh,e^c=eg,f^c=fh,g^c=gi

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, 53247);