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×#43648 |
| 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 | |
5392 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)(11,12,14,16)(13,15,17,18)(19,20)(21,23)(22,25)(24,26);(3,4)(5,6)(7,8)(9,10)(19,21,24,22)(20,25,26,23);(4,8)(5,9)(11,12)(13,15)(14,16)(17,18)(19,22,24,21)(20,25,26,23);(3,5,7,9)(4,6,8,10)(11,13,14,17)(12,15,16,18)(19,23,24,25)(20,21,26,22);(3,6)(4,9)(5,8)(7,10)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7574899);