Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#16768 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3450 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)(11,12,14,16)(13,17,18,15)(19,20)(21,25)(22,26)(23,24)(28,29)(30,34);(4,7)(8,10)(20,24)(21,26)(27,28,31,29)(30,32,34,33);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21,23,26)(20,25,24,22)(27,29,31,28)(30,32,34,33);(3,5,6,9)(4,8,7,10)(11,13,14,18)(12,15,16,17)(19,22,23,25)(20,21,24,26)(27,30)(28,32)(29,33)(31,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518700);