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×#62946 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
34024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 26 points:
(1,2);(4,6)(8,9)(12,16)(17,23)(18,24)(25,26);(3,4,7,6)(5,9,10,8);(3,5,7,10)(4,8,6,9)(11,12)(13,17)(14,18)(15,16)(19,25)(20,23)(21,26)(22,24);(3,6,7,4)(5,9,10,8)(11,13,15,20)(12,17,16,23)(14,21,22,19)(18,26,24,25);(11,14,15,22)(12,18,16,24)(13,19,20,21)(17,25,23,26);(11,12)(13,17)(14,18)(15,16)(19,25)(20,23)(21,26)(22,24)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493486);