#145393

order	512
nilpotency class	4
isoclinism	isoclinic class of #145359
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 324 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) = 128

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

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

Polycyclic presentation:

2^9|a^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=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fgi,g^b=gi,h^b=hi,e^c=eghi,f^c=fi

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, 52589);