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×#43239 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5694 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)(13,14)(15,18)(16,17)(19,20,22,24)(21,23,25,26);(3,4)(5,10)(6,7)(8,9)(12,17)(14,18);(4,7)(8,10)(11,13,16,15)(12,14,17,18)(19,20)(21,23)(22,24)(25,26);(3,4)(5,8)(6,7)(9,10)(11,14,16,18)(12,13,17,15)(19,21,22,25)(20,23,24,26);(3,5)(4,8)(6,9)(7,10)(11,15,16,13)(12,18,17,14)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7574397);