#123324

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	#493	D₄
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 228 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,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, 30520);