Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51936 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6112 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);(11,12)(13,16)(14,18)(15,17)(19,20,24,25)(21,26,23,22);(3,4,6,7)(5,9,10,8)(11,13)(12,14)(15,18)(16,17)(20,25)(21,23);(4,7)(8,9)(11,13,15,18)(12,14,17,16)(19,21,24,23)(20,26,25,22);(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,17)(15,18)(19,22,24,26)(20,21,25,23);(3,5,6,10)(4,8,7,9)(11,14,15,16)(12,13,17,18)(19,23,24,21)(20,26,25,22)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615087);