#126493

order	512
nilpotency class	4
isoclinism	isoclinic class of #125892
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1629
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

293 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,g^a=gi,d^b=dfi,e^b=eg,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi

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, 33689);