Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#44379 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(5,9)(7,10)(11,12,16,18)(13,14,17,15)(19,20,23,26)(21,24,22,25);(3,4)(5,10)(6,8)(7,9)(11,13,16,17)(12,15,18,14)(19,21,23,22)(20,24,26,25);(3,4,6,8)(5,7,9,10)(11,14,16,15)(12,17,18,13)(19,20)(21,24)(22,25)(23,26);(3,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,18)(16,17)(19,21,23,22)(20,24,26,25);(3,5)(4,7)(6,9)(8,10)(11,15,16,14)(12,13,18,17)(19,22,23,21)(20,25,26,24)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7575750);