#143833

order	512
nilpotency class	4
isoclinism	isoclinic class of #143807
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

381 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51029);