Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#23345 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,7)(5,9)(11,12)(13,15)(14,16)(17,18)(19,20,23,26,22,25,30,33)(21,27,29,34,28,24,31,32);(3,4,6,7)(5,10,9,8)(11,13)(12,15)(14,17)(16,18);(4,7)(8,10)(12,16)(13,17)(19,21)(20,24)(22,28)(23,29)(25,27)(26,32)(30,31)(33,34);(3,5,6,9)(4,8,7,10)(12,16)(15,18)(21,28)(24,27)(29,31)(32,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314728);