#140116

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=dg,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, 47312);