#145431

order	512
nilpotency class	4
isoclinism	isoclinic class of #145350
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#466
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

274 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=d,d^2=g,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fg,e^c=egh,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, 52627);