#145741

order	512
nilpotency class	4
isoclinism	isoclinic class of #145740
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

337 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(10,17)(11,23)(12,28)(13,14)(15,29)(19,32)(20,22)(21,30)(24,33)(25,39)(26,31)(27,37)(34,38)(36,40);(1,3)(2,5)(4,7)(6,8)(9,10,16,22)(11,24,27,36)(12,19,14,30)(13,20,28,17)(15,32,29,21)(18,31,35,25)(23,34,37,38)(26,40,39,33);(5,8)(9,11,12,25,29,39,13,23,16,27,14,31,15,26,28,37)(10,18,17,24,32,38,30,40,22,35,20,36,21,34,19,33)

Polycyclic presentation:

2^9|b^2=i,c^2=d,d^2=gi,e^2=g,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,f^c=fi

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, 52937);