#140168

order	512
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = 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	C₂³	#436
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

239 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=d,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,e^c=ei

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, 47364);