#144542

order	512
nilpotency class	4
isoclinism	isoclinic class of #144542
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10352
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#454
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 232 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) = 32
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=dg,c^2=g,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=di,e^c=egi,f^c=fh,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, 51738);