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×#51226 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5208 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)(13,18)(14,17)(15,16)(19,20,24,25)(21,26,29,33)(22,27,28,23)(30,31,32,34);(3,4,6,8)(5,7,10,9)(12,16)(17,18)(20,25)(21,29)(23,27)(30,32);(11,13,15,14)(12,17,16,18)(19,21)(20,26)(22,32)(23,31)(24,29)(25,33)(27,34)(28,30);(3,4)(5,7)(6,8)(9,10)(11,12)(13,17)(14,18)(15,16)(19,22,24,28)(20,27,25,23)(21,30,29,32)(26,31,33,34);(3,5,6,10)(4,7,8,9)(11,14,15,13)(12,18,16,17)(19,23,24,27)(20,28,25,22)(21,31,29,34)(26,30,33,32)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614468);