#104298

order	512
nilpotency class	3
isoclinism	isoclinic class of #104245
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	C₂⁴	#121
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

491 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,5)(4,7)(6,8)(9,10,15,12)(11,18,21,24)(13,23,14,17)(16,20,22,19)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,39,40,36);(3,8)(5,6)(9,11,15,21)(10,16,12,22)(13,20,14,19)(17,24,23,18)(25,27)(26,30)(28,33)(29,34)(31,36)(32,37)(35,40)(38,39);(2,5)(3,8)(6,7)(9,12,15,10)(11,18,21,24)(13,23,14,17)(16,19,22,20)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fi,c^b=cf,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,f^c=fg

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11494);