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×#23342 |
| 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)(19,20,24,25,23,28,21,27)(22,30,31,34,32,26,29,33);(3,4,6,7)(5,10,9,8)(11,13)(12,15)(14,17)(16,18)(19,21,23,24)(20,25,28,27)(22,31,32,29)(26,34,30,33);(4,7)(8,10)(12,16)(13,17)(19,22)(20,26)(21,29)(23,32)(24,31)(25,33)(27,34)(28,30);(3,5,6,9)(4,8,7,10)(12,16)(15,18)(19,21,23,24)(20,27,28,25)(22,31,32,29)(26,33,30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314729);