Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#29976 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1898 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)(8,10)(11,12,14,16)(13,17,18,15)(19,20)(21,28)(22,32)(23,27)(24,26)(25,31)(29,34)(30,33);(3,4,6,7)(5,8,9,10)(19,21,23,31)(20,25,27,28)(22,30,24,29)(26,33,32,34);(3,5)(4,8)(6,9)(7,10)(11,13,14,18)(12,15,16,17)(21,29)(22,24)(25,33)(26,32)(28,34)(30,31);(5,9)(8,10)(19,22,23,24)(20,26,27,32)(21,30,31,29)(25,33,28,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340497);