#145392

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	C₂×#493	C₂²
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

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

normal subgroups only

This group has 334 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)(10,17)(11,22)(12,24)(13,14)(15,30)(18,35)(19,20)(21,31)(23,38)(25,28)(26,27)(29,33)(32,39)(36,37);(2,5)(6,7)(9,10,16,21)(11,23,28,38)(12,29,13,19)(14,31,24,17)(15,33,30,20)(18,32,37,26)(22,35,25,36)(27,34,39,40);(2,6)(4,8)(5,7)(9,11,24,26,15,27,13,22,16,28,14,32,30,39,12,25)(10,18,29,23,20,36,31,40,21,37,19,38,33,35,17,34)

Polycyclic presentation:

2^9|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=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, 52588);