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×#56761 |
| 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 | |
4310 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);(4,8)(5,11)(10,15)(13,17)(19,20,24,22)(21,25,27,29)(23,30,31,26)(28,34,33,32)(35,36)(37,42)(38,43)(39,47)(40,41)(44,48)(45,46)(49,50);(6,14)(10,15)(13,17)(16,18)(20,22)(21,27)(23,31)(32,34)(36,41)(38,42)(44,47)(46,49);(3,4,7,8)(5,12,11,9)(6,10,14,15)(13,18,17,16)(19,21)(20,25)(22,29)(23,28)(24,27)(26,32)(30,34)(31,33)(35,37,40,43)(36,42,41,38)(39,45,48,50)(44,49,47,46);(3,5,7,11)(4,9,8,12)(6,13,14,17)(10,16,15,18)(19,22,24,20)(21,25,27,29)(23,30,31,26)(28,32,33,34)(35,38,40,42)(36,43,41,37)(39,46,48,49)(44,50,47,45);(3,6)(4,10)(5,13)(7,14)(8,15)(9,16)(11,17)(12,18)(19,23,24,31)(20,26,22,30)(21,28,27,33)(25,32,29,34)(35,39,40,48)(36,44,41,47)(37,45,43,50)(38,46,42,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006718);