#123308

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

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

normal subgroups only

This group has 231 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) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,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, 30504);