#145878

order	512
nilpotency class	4
isoclinism	isoclinic class of #143259
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1624
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10577
prime radical	C₂³	#456
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

225 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dg,c^2=dg,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=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, 53074);