Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11495 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C22×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,11,13,8)(6,9)(7,10)(12,17,18,15)(14,16)(19,20,23,29,25,26,32,33,24,28,31,34,21,30,22,27);(3,5,7,13)(4,8,10,11)(6,12,14,18)(9,15,16,17)(19,21,24,25)(20,26,28,30)(22,23,32,31)(27,34,33,29);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,22,21,31,24,32,25,23)(20,27,30,34,28,33,26,29)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387358);