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×#50548 |
| 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 | |
5402 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)(5,9)(6,8)(7,10)(11,12)(13,14)(15,16)(17,18)(19,20,23,24)(21,27,28,25)(22,26,31,32)(29,34,30,33);(3,4,6,8)(5,7,10,9)(12,16)(14,17)(20,24)(21,28)(26,32)(29,30);(11,13)(12,17)(14,16)(15,18)(19,21,23,28)(20,25,24,27)(22,30,31,29)(26,34,32,33);(3,4)(5,7)(6,8)(9,10)(11,14,15,17)(12,13,16,18)(19,22,23,31)(20,26,24,32)(21,29,28,30)(25,33,27,34);(3,5,6,10)(4,7,8,9)(11,13)(12,17)(14,16)(15,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,33)(30,34)(31,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613204);