#153266

order	512
nilpotency class	5
isoclinism	isoclinic class of #153078
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

313 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60462);