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×#36057 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(4,8)(5,10)(11,12,15,14)(13,17,18,16)(19,20,24,25)(21,23,28,26)(22,27,31,33)(29,32,34,30);(3,4)(5,9)(6,10)(7,8)(11,13,15,18)(12,16,14,17)(20,25)(21,28)(27,33)(29,34);(3,5,7,10)(4,9,8,6)(11,12,15,14)(13,16,18,17)(19,21,24,28)(20,26,25,23)(22,29,31,34)(27,30,33,32);(11,12,15,14)(13,16,18,17)(19,22)(20,27)(21,29)(23,32)(24,31)(25,33)(26,30)(28,34);(3,6,7,9)(4,10,8,5)(11,14,15,12)(13,17,18,16)(19,23,24,26)(20,28,25,21)(22,30,31,32)(27,29,33,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536722);