Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11857 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,11,13,8)(6,9)(7,10)(12,17,14,16)(15,18)(20,26,28,31)(21,22)(23,24)(27,29,34,30)(32,33);(3,5)(4,8)(6,14)(7,13)(9,17)(10,11)(12,15)(16,18)(19,20,25,28)(21,31,33,26)(22,34,32,27)(23,30,24,29);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,21,24,22,25,33,23,32)(20,27,30,26,28,34,29,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389405);