Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62857 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
34446 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)(13,17)(14,21)(15,16)(18,22)(19,23)(20,26)(24,25);(3,4)(5,8)(6,7)(9,10)(13,19)(17,23)(20,25)(24,26);(12,16)(17,23)(18,21)(24,26);(11,12)(13,17)(14,18)(15,16)(19,23)(20,24)(21,22)(25,26);(11,13,15,19)(12,17,16,23)(14,20,22,25)(18,24,21,26);(3,5,6,9)(4,8,7,10)(11,14)(12,18)(13,20)(15,22)(16,21)(17,24)(19,25)(23,26)164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493209);