Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9342 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1284 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)(5,8,12,16)(6,13,14,9)(7,10)(11,15)(17,18)(19,20,23,27,24,28,22,26)(21,29,31,25,32,33,30,34);(3,5)(4,8,10,16)(6,11)(7,12)(9,15,13,18)(14,17)(19,21)(20,25)(22,31)(23,30)(24,32)(26,33)(27,29)(28,34);(3,6,7,14)(4,9,10,13)(5,11,12,17)(8,15,16,18)(19,21)(20,25)(22,30)(23,31)(24,32)(26,29)(27,33)(28,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106913);