#145890

order	512
nilpotency class	4
isoclinism	isoclinic class of #145864
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2058
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#450
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

262 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dh,c^2=dg,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=eg,f^c=fh,g^c=gi

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, 53086);