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×#59670 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,16,14)(13,18,17,15)(19,20,24,21)(22,26,23,25);(4,7)(8,9)(11,13)(12,15)(14,18)(16,17)(19,21,24,20)(22,25,23,26);(11,14,16,12)(13,18,17,15)(19,22,24,23)(20,25,21,26);(3,4)(5,8)(6,7)(9,10)(19,23,24,22)(20,26,21,25);(3,5,6,10)(4,8,7,9)(11,15,16,18)(12,17,14,13)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573440);