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×#36058 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,7,10)(5,8,6,9)(12,17)(13,18)(19,20,24,25)(21,22,27,26)(23,28,32,33)(29,30,34,31);(3,5,7,6)(4,8,10,9)(11,12)(13,14)(15,18)(16,17)(20,25)(21,27)(28,33)(29,34);(3,4)(5,8)(6,9)(7,10)(11,13,16,18)(12,14,17,15)(19,21,24,27)(20,26,25,22)(23,29,32,34)(28,31,33,30);(3,5,7,6)(4,8,10,9)(11,14,16,15)(12,13,17,18)(19,22,24,26)(20,27,25,21)(23,31,32,30)(28,29,33,34);(3,6,7,5)(4,9,10,8)(11,15,16,14)(12,18,17,13)(19,23,24,32)(20,28,25,33)(21,29,27,34)(22,30,26,31)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536704);