#104803

order	512
nilpotency class	3
isoclinism	isoclinic class of #104789
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

426 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 80 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)(9,10,12,15)(11,18)(13,17,14,22)(16,19)(20,23)(21,24)(25,26)(27,30)(28,35,37,31)(29,32)(33,38,36,40)(34,39);(2,5)(6,7)(10,15)(11,19,21,20)(16,24,23,18)(17,22)(25,27)(26,30)(28,36,37,33)(29,34)(31,40,35,38)(32,39);(2,6)(4,8)(5,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,24)(18,22)(25,28,29,37)(26,31,32,35)(27,33,34,36)(30,38,39,40)

Polycyclic presentation:

2^9|a^2=d,b^2=dgh,c^2=i,e^2=gh,f^2=gi,b^a=bd,c^a=ce,e^a=eh,f^a=fgi,c^b=cf,e^b=ei,f^b=fi,d^c=dg,e^c=egh,f^c=fgi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11999);