#144536

order	512
nilpotency class	4
isoclinism	isoclinic class of #143259
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1624
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10351
prime radical	#3147	C₂²
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

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

normal subgroups only

This group has 277 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:

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

Polycyclic presentation:

2^9|a^2=d,b^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, 51732);