#123291

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	#19971	C₂
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

259 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 52 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)(17,18,22,27)(19,28,31,46)(20,24,35,42)(21,36,34,26)(23,38,40,29)(25,43,41,33)(30,47,37,44)(32,45,48,39);(2,6)(3,10)(4,11)(7,8)(9,15)(12,16)(13,14)(17,19,22,31)(18,23,27,40)(20,32,35,48)(21,37,34,30)(24,28,42,46)(25,44,41,47)(26,45,36,39)(29,43,38,33);(18,24)(20,33)(21,34)(23,28)(25,26)(27,42)(29,32)(30,37)(35,43)(36,41)(38,48)(39,47)(40,46)(44,45)

Polycyclic presentation:

2^9|a^2=h,b^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, 30487);