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×#56166 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4148 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)(5,9)(6,13)(7,8)(10,14)(11,17)(12,15)(16,18)(19,20,24,23)(21,27,28,25)(22,30,31,26)(29,32,34,33)(36,37)(38,45)(39,47)(48,49);(4,8)(9,15)(10,13)(16,17)(21,28)(25,27)(29,34)(32,33)(35,36,40,37)(38,43,45,41)(39,44,47,42)(46,48,50,49);(3,5,7,12)(4,9,8,15)(6,11,14,18)(10,16,13,17)(19,21)(20,25)(22,29)(23,27)(24,28)(26,32)(30,33)(31,34)(35,37,40,36)(38,41,45,43)(39,44,47,42)(46,48,50,49);(3,4)(5,9)(6,10)(7,8)(11,16)(12,15)(13,14)(17,18)(19,22,24,31)(20,26,23,30)(21,29,28,34)(25,32,27,33)(35,38,40,45)(36,41,37,43)(39,46,47,50)(42,48,44,49);(3,6)(4,10)(5,11)(7,14)(8,13)(9,16)(12,18)(15,17)(19,23,24,20)(21,27,28,25)(22,30,31,26)(29,33,34,32)(35,39,40,47)(36,42,37,44)(38,46,45,50)(41,48,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010208);