Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#45573 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5098 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);(5,9)(8,10)(13,17)(16,18)(19,20,24,25)(21,28,26,22)(23,31,32,27)(29,30,33,34);(4,7)(5,9)(11,12,15,14)(13,16,17,18)(20,25)(21,26)(23,32)(30,34);(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,18)(15,17)(19,21,24,26)(20,22,25,28)(23,29,32,33)(27,30,31,34);(3,5)(4,8)(6,9)(7,10)(11,14,15,12)(13,18,17,16)(19,22)(20,26)(21,25)(23,30)(24,28)(27,33)(29,31)(32,34);(3,4)(5,8)(6,7)(9,10)(19,23)(20,27)(21,29)(22,30)(24,32)(25,31)(26,33)(28,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609350);