#153211

order	512
nilpotency class	5
isoclinism	isoclinic class of #153090
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

265 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,e^b=ei,e^c=eg,g^c=gh,h^c=hi

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, 60407);