#145864

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

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

normal subgroups only

This group has 294 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,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, 53060);