#146070

order	512
nilpotency class	4
isoclinism	isoclinic class of #146048
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
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

238 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=egh,c^2=d,d^2=i,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,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, 53266);