#131366

order	512
nilpotency class	4
isoclinism	isoclinic class of #131280
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

662 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 100 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)(3,4,8,6)(5,7)(9,10,13,19,14,20,27,34,15,17,29,35,28,36,12,18)(11,21,24,31,25,37,22,38,23,39,26,16,30,32,40,33);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,26)(13,22)(14,30)(15,23)(17,31)(18,37)(19,21)(20,38)(24,27)(25,28)(29,40)(32,34)(33,36)(35,39);(1,4,5,6)(2,3,7,8)(10,17)(12,27)(13,29)(16,31)(20,36)(22,40)(24,26)(33,38)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,f^c=fgi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38562);