#102487

order	512
nilpotency class	3
isoclinism	isoclinic class of #102469
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#2856	C₂²
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 507 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dh,e^c=ei,f^c=fg

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9683);