Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52792 |
| 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 | |
4294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4)(5,11)(6,14)(7,8)(9,12)(10,15)(13,16)(17,18)(19,20,24,25)(21,26,27,22)(23,31,32,28)(29,30,34,33)(38,46)(41,48)(44,50)(47,49);(4,8)(5,12)(10,14)(13,18)(20,25)(22,26)(28,31)(30,33)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,5)(4,9)(6,13)(7,12)(8,11)(10,16)(14,17)(15,18)(19,21)(20,26)(22,25)(23,29)(24,27)(28,33)(30,31)(32,34)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,48)(47,50);(3,4)(5,11)(6,10)(7,8)(9,12)(13,17)(14,15)(16,18)(19,22,24,26)(20,27,25,21)(23,30,32,33)(28,34,31,29)(35,37)(36,40)(38,44)(39,45)(41,47)(42,43)(46,50)(48,49);(3,6,7,15)(4,10,8,14)(5,13,12,18)(9,16,11,17)(19,23,24,32)(20,28,25,31)(21,29,27,34)(22,30,26,33)(35,38)(36,41)(37,44)(39,46)(40,47)(42,48)(43,49)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999728);