#145198

order	512
nilpotency class	4
isoclinism	isoclinic class of #145143
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
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 ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 332 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=dgh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,c^b=cf,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=eg,f^c=fhi,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, 52394);