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×#53811 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5190 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)(13,18)(14,17)(15,16)(20,24)(21,27)(22,30)(32,34);(3,4)(5,9)(6,7)(8,10)(11,13)(12,17)(14,16)(15,18)(19,20,23,24)(21,28,27,25)(22,26,30,31)(29,34,33,32);(4,7)(8,9)(13,14)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(33,34);(3,4)(5,8)(6,7)(9,10)(11,14,16,13)(12,17,15,18)(19,21,23,27)(20,25,24,28)(22,29,30,33)(26,32,31,34);(3,5)(4,8)(6,10)(7,9)(11,15,16,12)(13,17,14,18)(19,22)(20,26)(21,29)(23,30)(24,31)(25,32)(27,33)(28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010572);