#140104

order	512
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1923
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

268 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,5)(4,7)(6,8)(9,10,15,21,14,19,12,30,16,22,31,18,13,20,24,37)(11,23,28,39,27,36,25,33,29,34,40,38,26,17,32,35);(9,11,24,32,13,26,31,40,16,29,12,25,14,27,15,28)(10,17,30,39,20,34,21,35,22,36,37,38,19,23,18,33);(2,5)(3,8)(6,7)(9,12,13,15,16,24,14,31)(10,18,19,37,22,21,20,30)(11,25,26,28,29,32,27,40)(17,33,23,38,36,35,34,39)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=d,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,e^c=egi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47300);