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×#55198 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5022 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);(11,12)(13,16)(14,15)(17,18)(19,20,24,25)(21,23,29,26)(22,30,31,27)(28,34,32,33);(3,4,7,8)(5,6,9,10)(12,14)(13,17)(20,25)(22,31)(23,26)(28,32);(4,8)(5,9)(11,13,15,17)(12,16,14,18)(19,21)(20,26)(22,32)(23,25)(24,29)(27,33)(28,31)(30,34);(3,5)(4,6)(7,9)(8,10)(11,13)(12,16)(14,18)(15,17)(19,22,24,31)(20,27,25,30)(21,28,29,32)(23,33,26,34);(3,6)(4,5)(7,10)(8,9)(11,14,15,12)(13,18,17,16)(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, 8009543);