Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10326 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10332 |
| 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 | |
1668 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);(3,4,7,11)(5,9,13,18)(6,8,14,10)(12,17,15,16)(19,20)(21,24)(22,25)(23,26)(27,32)(28,31)(29,33)(30,34);(4,8)(6,14)(9,17)(10,11)(12,15)(16,18)(19,21)(20,24)(22,27)(23,28)(25,32)(26,31)(29,34)(30,33);(3,5,7,13)(4,9,11,18)(6,15,14,12)(8,16,10,17)(19,22,23,29)(20,25,26,33)(21,27,28,34)(24,30,31,32)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141554);