#144540

order	512
nilpotency class	4
isoclinism	isoclinic class of #143329
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1634
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10352
prime radical	C₂³	#454
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 230 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dg,c^2=g,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=eg,f^c=fhi,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, 51736);