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×#62917 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
34836 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)(11,12,18,21)(13,22,16,19)(14,17,15,20)(23,24,25,26);(11,13,18,16)(12,19,21,22)(14,25,15,23)(17,24,20,26);(3,4)(5,8)(6,7)(9,10)(11,14,18,15)(12,20,21,17)(13,23,16,25)(19,24,22,26);(5,9)(8,10)(11,15,18,14)(12,17,21,20)(13,23,16,25)(19,24,22,26);(3,5,6,9)(4,8,7,10)(11,16,18,13)(12,19,21,22)(14,25,15,23)(17,26,20,24);(3,4)(5,8)(6,7)(9,10)(11,17)(12,14)(13,24)(15,21)(16,26)(18,20)(19,25)(22,23)146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493455);