Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#54992 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(19,20,24,25)(21,23,29,26)(22,30,31,27)(28,34,32,33);(6,10)(8,9)(11,12,15,16)(13,14,17,18)(20,25)(22,31)(23,26)(28,32);(4,5)(8,9)(12,16)(13,17)(19,21)(20,26)(22,32)(23,25)(24,29)(27,33)(28,31)(30,34);(3,5,7,4)(6,9,10,8)(11,13)(12,14)(15,17)(16,18)(19,22,24,31)(20,27,25,30)(21,28,29,32)(23,33,26,34);(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,18)(16,17)(19,23,24,26)(20,21,25,29)(22,33,31,34)(27,28,30,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009538);