#144541

order	512
nilpotency class	4
isoclinism	isoclinic class of #143326
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

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

normal subgroups only

This group has 234 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dg,c^2=g,d^2=i,e^2=g,f^2=h,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=fhi,g^b=gi,h^b=hi,e^c=eg,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51737);