#123401

order	512
nilpotency class	4
isoclinism	isoclinic class of #123383
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
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

156 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,8)(3,10,11,15)(4,13,14,16)(6,12,7,9)(17,18)(19,22)(20,23)(21,29)(24,28)(25,27)(26,30)(31,32);(1,3,10,8,6,13,4,12,5,11,15,2,7,16,14,9)(18,22)(23,30)(24,31)(25,29);(2,8)(3,11)(9,12)(13,16)(18,23)(20,28)(21,27)(22,30)(24,25)(26,32)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=dfhi,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,c^b=ch,e^c=eg,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30597);