#145465

order	512
nilpotency class	4
isoclinism	isoclinic class of #145365
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#463
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 282 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=dh,d^2=gi,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,g^b=gi,h^b=hi,e^c=eghi,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52661);