#123293

order	512
nilpotency class	4
isoclinism	isoclinic class of #123275
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	C₂×D₈	D₈
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

255 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9)(3,6,12,15)(4,13,11,8)(7,16,14,10)(17,18,23,28)(19,29,20,32)(21,35,40,48)(22,41,38,27)(24,33,25,30)(26,44,47,39)(31,46,34,45)(36,43,37,42);(2,6)(3,10)(4,11)(7,8)(9,15)(12,16)(13,14)(17,19,23,20)(18,24,28,25)(21,36,40,37)(22,34,38,31)(26,45,47,46)(27,43,41,42)(29,48,32,35)(30,44,33,39);(17,20,23,19)(18,25,28,24)(21,37,40,36)(22,31,38,34)(26,46,47,45)(27,42,41,43)(29,35,32,48)(30,39,33,44)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi,d^b=df,e^b=eg,f^b=fh,g^b=gi,d^c=dg,e^c=eg,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30489);