#145189

order	512
nilpotency class	4
isoclinism	isoclinic class of #145145
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
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 ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

314 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=dh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,c^b=cf,f^b=fh,h^b=hi,d^c=di,e^c=eg,f^c=fhi,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, 52385);