Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#9984 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,8)(6,9)(7,10)(11,14)(12,15)(13,17)(16,18)(19,20,24,27)(21,28,30,33)(22,32,23,26)(25,31,34,29);(3,5,7,12)(4,8,10,15)(6,11,13,18)(9,16,17,14)(19,21,24,30)(20,25,27,34)(22,31,23,29)(26,33,32,28);(3,6)(4,9)(5,11)(7,13)(8,14)(10,17)(12,18)(15,16)(21,29)(22,23)(25,33)(26,32)(28,34)(30,31)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139706);