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×#62913 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
34088 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)(12,16)(14,22)(17,19)(20,26);(11,12,15,16)(13,19,21,17)(14,18,22,23)(20,25,26,24);(3,4)(5,8)(6,7)(9,10)(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25);(5,9)(8,10)(11,14)(12,18)(13,20)(15,22)(16,23)(17,24)(19,25)(21,26);(3,5,6,9)(4,8,7,10);(3,4)(5,8)(6,7)(9,10)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493451);