#153343

order	512
nilpotency class	5
isoclinism	isoclinic class of #153078
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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

222 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 78 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)(10,17)(11,22)(12,13)(14,26)(15,32)(18,28)(19,35)(20,37)(21,25)(23,24)(27,36)(29,30)(31,39)(33,40)(34,38);(5,6)(7,8)(9,10,18,24,14,20,27,34,32,38,30,37,40,23,12,17,16,22,13,19,33,39,29,25,15,21,36,31,26,35,28,11);(2,4)(6,8)(9,11,16,17)(10,12,22,28)(13,23,18,35)(14,31,33,37)(15,34,32,25)(19,26,24,40)(20,30,39,36)(21,29,38,27)

Polycyclic presentation:

2^9|a^2=de,b^2=dgi,c^2=i,d^2=gh,e^2=g,f^2=g,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,h^a=hi,c^b=cf,d^c=dg,e^c=egh,f^c=fgh,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, 60539);