#123325

order	512
nilpotency class	4
isoclinism	isoclinic class of #123311
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	#20308	C₂
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

240 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30521);