#144537

order	512
nilpotency class	4
isoclinism	isoclinic class of #143264
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

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

normal subgroups only

This group has 273 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dg,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=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, 51733);