#140072

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×D₈
prime radical	C₂³	#443
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

215 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 215 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

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, 47268);