#123321

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, prime radical	C₂²	#1619
commutator, Frattini, cosocle	C₈²	C₂³
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 216 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) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,11)(3,12,15,30)(4,9,6,21)(5,17,20,28)(8,23,25,14)(10,16,29,19)(13,31,22,27)(18,24,32,26)(33,34,40,45)(35,46,36,51)(37,52,56,62)(38,44,39,43)(41,48,59,47)(42,55,63,54)(49,61,50,60)(53,58,64,57);(1,3)(2,8)(4,13)(5,18)(6,22)(7,15)(9,24)(10,27)(11,25)(12,17)(14,19)(16,23)(20,32)(21,26)(28,30)(29,31)(33,35,40,36)(34,41,45,59)(37,53,56,64)(38,50,39,49)(42,60,63,61)(43,58,44,57)(46,62,51,52)(47,55,48,54);(1,4,7,6)(2,9,11,21)(3,13,15,22)(5,19,20,16)(8,24,25,26)(10,28,29,17)(12,27,30,31)(14,32,23,18)(33,36,40,35)(34,41,45,59)(37,53,56,64)(38,49,39,50)(42,61,63,60)(43,58,44,57)(46,52,51,62)(47,55,48,54)

Polycyclic presentation:

2^9|a^2=h,b^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, 30517);