#144753

order	512
nilpotency class	4
isoclinism	isoclinic class of #144716
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

355 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=d,d^2=i,e^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,f^b=fi,e^c=egi,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, 51949);