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×#52523 |
| 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 | |
4534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(5,9)(8,10)(11,12,15,16)(13,17,19,20)(14,22,23,18)(21,26,25,24)(27,28,32,35)(29,33,37,31)(30,38,41,34)(36,40,39,42);(3,4)(5,8)(6,7)(9,10)(12,16)(13,19)(18,22)(21,25)(27,29,32,37)(28,31,35,33)(30,39,41,36)(34,42,38,40);(4,7)(8,10)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,29)(28,33)(30,36)(31,35)(32,37)(34,42)(38,40)(39,41);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,26)(23,25)(27,30,32,41)(28,34,35,38)(29,36,37,39)(31,40,33,42);(3,5)(4,8)(6,9)(7,10)(11,14,15,23)(12,18,16,22)(13,21,19,25)(17,24,20,26)(27,31,32,33)(28,29,35,37)(30,40,41,42)(34,36,38,39)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999541);