Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#59730 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5698 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);(3,4)(5,6)(7,8)(9,10)(11,12,15,14)(13,17,18,16)(19,20,24,23)(21,22,26,25);(4,8)(5,9)(11,13)(12,16)(14,17)(15,18)(21,26)(22,25);(11,12,15,14)(13,16,18,17)(19,21,24,26)(20,25,23,22);(3,5,7,9)(4,6,8,10)(11,14,15,12)(13,17,18,16)(19,22)(20,21)(23,26)(24,25);(3,6)(4,9)(5,8)(7,10)(11,13)(12,16)(14,17)(15,18)(19,23,24,20)(21,22,26,25)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573572);