#145376

order	512
nilpotency class	4
isoclinism	isoclinic class of #145376
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19189	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

321 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

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

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, 52572);