#168486

order	512
nilpotency class	3
isoclinism	isoclinic class of #168113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

717 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75682);