#123355

order	512
nilpotency class	4
isoclinism	isoclinic class of #123331
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1619
commutator, Frattini, cosocle	C₈²	C₂³
prime radical	#19884	C₂
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

278 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=fh,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=egi,f^a=fh,g^a=gi,d^b=df,e^b=eg,f^b=fh,g^b=gi,d^c=dg,e^c=egi,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, 30551);