Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #1667 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,8)(6,7)(9,10,15,19)(11,16,14,29)(12,22,24,39)(13,25,28,38)(17,32,34,23)(18,21,36,27)(20,37,30,35)(26,31,40,33);(1,2,4,7)(3,6,8,5)(9,11,15,14)(10,16,19,29)(12,20,24,30)(13,21,28,27)(17,31,34,33)(18,25,36,38)(22,37,39,35)(23,26,32,40);(9,12,15,24)(10,17,19,34)(11,20,14,30)(13,26,28,40)(16,31,29,33)(18,35,36,37)(21,32,27,23)(22,38,39,25);(2,6)(3,8)(5,7)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309719 |
| maximal normal subgroup of | #72309719 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398986);