#349321

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

747 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 747 conjugacy classes of subgroups — too many to draw. Only its 164 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 — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 256517);