Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#35932 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7316 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);(11,12)(13,18)(14,16)(15,17)(19,20,25,26)(21,22,24,23);(3,4)(5,8)(6,9)(7,10)(11,12)(13,18)(14,16)(15,17)(19,21,25,24)(20,23,26,22);(3,5)(4,7)(6,10)(8,9)(11,13,15,14)(12,16,17,18)(19,22,25,23)(20,21,26,24);(3,5)(4,8)(6,10)(7,9)(11,14,15,13)(12,16,17,18)(19,23,25,22)(20,21,26,24);(3,4)(5,7)(6,9)(8,10)(11,12)(13,16)(14,18)(15,17)(19,24,25,21)(20,23,26,22)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536557);