#134594

order	512
nilpotency class	4
isoclinism	isoclinic class of #134444
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#417
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

300 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=g,d^2=i,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,e^c=eh,f^c=fgh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41790);