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×#62922 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
34418 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)(13,19)(14,17)(15,18)(16,21)(20,25)(22,24)(23,26);(11,12,15,18)(13,16,21,19)(14,17,22,24)(20,23,26,25);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,17)(15,18)(19,21)(20,23)(22,24)(25,26);(5,9)(8,10)(11,13)(12,16)(14,20)(15,21)(17,23)(18,19)(22,26)(24,25);(3,5,6,9)(4,8,7,10);(3,4)(5,8)(6,7)(9,10)(11,14,15,22)(12,17,18,24)(13,20,21,26)(16,23,19,25)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493460);