#146197

order	512
nilpotency class	4
isoclinism	isoclinic class of #146196
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	Q₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

249 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=degi,d^2=g,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,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, 53393);