#146165

order	512
nilpotency class	4
isoclinism	isoclinic class of #143943
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#459
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

244 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=egi,b^2=d,c^2=di,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fh,h^b=hi,e^c=eg,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53361);