#153341

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

Polycyclic presentation:

2^9|a^2=de,b^2=dg,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, 60537);