Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#16208 |
| 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 | |
3702 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);(5,9)(8,10)(12,15)(13,17)(19,20)(21,25)(22,23)(24,26)(28,29)(30,34);(4,7)(8,10)(12,15)(16,18)(20,23)(21,26)(27,28,31,29)(30,33,34,32);(3,4,6,7)(5,8,9,10)(11,12,14,15)(13,16,17,18)(19,20)(21,24)(22,23)(25,26)(27,29,31,28)(30,33,34,32);(3,5)(4,8)(6,9)(7,10)(11,13,14,17)(12,16,15,18)(19,21,22,26)(20,24,23,25)(27,30)(28,32)(29,33)(31,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 517569);