#146285

order	512
nilpotency class	4
isoclinism	isoclinic class of #144579
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#486
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2099
prime radical	C₂³	#474
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=di,b^2=de,c^2=d,d^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,f^c=fgh,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, 53481);