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×#52882 |
| 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 | |
4230 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)(5,11,12,8)(6,9,14,16)(13,17,18,15)(19,20,24,22)(21,27,28,25)(23,30,31,26)(29,32,34,33)(38,46)(41,48)(44,50)(47,49);(5,12)(8,11)(13,18)(15,17)(20,22)(21,28)(26,30)(29,34)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,5)(4,8)(6,13)(7,12)(9,15)(10,11)(14,18)(16,17)(19,21,24,28)(20,25,22,27)(23,29,31,34)(26,32,30,33)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,48)(47,50);(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,22,24,20)(21,25,28,27)(23,30,31,26)(29,32,34,33)(35,37)(36,40)(38,44)(39,45)(41,47)(42,43)(46,50)(48,49);(3,6,7,14)(4,9,10,16)(5,13,12,18)(8,15,11,17)(19,23,24,31)(20,26,22,30)(21,29,28,34)(25,32,27,33)(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, 7999755);