Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#7980 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #95 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1750 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)(5,13)(6,10)(7,11)(8,12)(9,14)(15,18)(16,17)(19,20)(21,27,29,24)(22,25)(23,26)(28,33,34,31)(30,32);(3,5,7,15,8,16,6,14)(4,9,10,17,12,18,11,13)(19,21)(20,24)(22,28)(23,29)(25,31)(26,27)(30,34)(32,33);(3,6,8,7)(4,10,12,11)(5,14,16,15)(9,17,18,13)(19,22,23,30)(20,25,26,32)(21,28,29,34)(24,31,27,33)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88429);