#123289

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, 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

236 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^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, 30485);