Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52556 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(5,9)(8,10)(11,12,15,13)(14,17,18,16)(19,20,23,25)(21,26,24,22);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,26);(4,7)(8,10)(12,13)(16,17)(19,20,23,25)(21,22,24,26);(3,4)(5,8)(6,7)(9,10)(11,13,15,12)(14,17,18,16)(19,20)(21,26)(22,24)(23,25);(3,5)(4,8)(6,9)(7,10)(11,14,15,18)(12,16,13,17)(19,22,23,26)(20,24,25,21)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999527);