Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62921 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
34494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 26 points:
(1,2);(4,7)(8,10)(11,12,16,17)(13,14,19,22)(15,18,23,24)(20,21,25,26);(12,17)(13,19)(18,24)(20,25);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,20)(16,19)(17,22)(18,21)(23,25)(24,26);(5,9)(8,10)(11,14)(12,13)(15,21)(16,22)(17,19)(18,20)(23,26)(24,25);(3,5,6,9)(4,8,7,10);(3,4)(5,8)(6,7)(9,10)(11,15,16,23)(12,18,17,24)(13,20,19,25)(14,21,22,26)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493459);