#104720

order	512
nilpotency class	3
isoclinism	isoclinic class of #102905
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

693 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 693 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=dg,c^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fi,e^c=eg,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11916);