#144927

order	512
nilpotency class	4
isoclinism	isoclinic class of #144892
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
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×#336 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

415 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=dg,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,d^c=di,e^c=egi,f^c=fg,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52123);