Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9056 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #386 |
| prime radical | C24 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1138 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);(4,8,10,13)(5,11,12,14)(19,20)(21,29,32,33)(22,26,25,31)(23,27)(24,28)(30,34);(3,4,6,13,7,10,15,8)(5,9,14,17,12,18,11,16)(19,21,30,31,24,32,23,26)(20,25,34,29,28,22,27,33);(3,5)(4,9)(6,14)(7,12)(8,16)(10,18)(11,15)(13,17)(19,22,24,25)(20,26,28,31)(21,27,32,34)(23,29,30,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102669);