#145992

order	512
nilpotency class	4
isoclinism	isoclinic class of #145984
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2067
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#474
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

219 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=de,b^2=d,c^2=g,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,f^b=fg,d^c=dg,e^c=eh,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53188);