Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#58531 |
| 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 | |
4300 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);(3,4,7,10)(5,11,13,8)(6,9,14,16)(12,17,18,15)(19,20,24,22)(21,25,29,27)(23,30,31,26)(28,33,34,32)(35,36)(37,41)(38,45)(39,42)(40,43)(44,48)(46,50)(47,49);(6,14)(9,16)(12,18)(15,17)(20,22)(25,27)(26,30)(32,33)(35,37,40,45)(36,41,43,38)(39,44,47,50)(42,48,49,46);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,21,24,29)(20,25,22,27)(23,28,31,34)(26,32,30,33)(35,37)(36,38)(39,44)(40,45)(41,43)(42,46)(47,50)(48,49);(3,5,7,13)(4,8,10,11)(6,12,14,18)(9,15,16,17)(19,22,24,20)(21,27,29,25)(23,30,31,26)(28,33,34,32)(35,38)(36,37)(39,46)(40,41)(42,44)(43,45)(47,48)(49,50);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,23,24,31)(20,26,22,30)(21,28,29,34)(25,32,27,33)(35,39,40,47)(36,42,43,49)(37,44,45,50)(38,46,41,48)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011452);