#145873

order	512
nilpotency class	4
isoclinism	isoclinic class of #145864
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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

286 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=dh,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=di,e^c=eg,f^c=fh,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, 53069);