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×#54338 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(4,7)(5,9)(11,12)(13,17)(14,16)(15,18)(19,20,24,21)(22,25,29,27)(23,28,31,26)(30,33,34,32);(3,4,6,7)(5,8,9,10)(11,13,16,15)(12,17,14,18)(20,21)(22,29)(23,31)(32,33);(3,4)(5,8)(6,7)(9,10)(11,14,16,12)(13,17,15,18)(19,21,24,20)(22,25,29,27)(23,28,31,26)(30,32,34,33);(3,4)(5,8)(6,7)(9,10)(11,15,16,13)(12,17,14,18)(19,22,24,29)(20,25,21,27)(23,30,31,34)(26,32,28,33);(3,5)(4,8)(6,9)(7,10)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010700);