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×#52794 |
| 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);(3,4)(5,9)(6,7)(8,10)(11,12)(13,19)(14,18)(15,16)(17,21)(20,25)(22,23)(24,26)(27,28,32,33)(29,34,36,31)(30,38,39,35)(37,40,42,41);(3,4,6,7)(5,8,10,9)(12,16)(17,19)(18,23)(24,25)(28,33)(29,36)(30,39)(40,41);(4,7)(8,9)(11,13)(12,17)(14,20)(15,21)(16,19)(18,24)(22,26)(23,25)(27,29)(28,34)(30,37)(31,33)(32,36)(35,41)(38,40)(39,42);(3,4)(5,8)(6,7)(9,10)(11,12)(13,17)(14,18)(15,16)(19,21)(20,24)(22,23)(25,26)(27,30,32,39)(28,35,33,38)(29,37,36,42)(31,40,34,41);(3,5,6,10)(4,8,7,9)(11,14,15,22)(12,18,16,23)(13,20,21,26)(17,24,19,25)(27,31,32,34)(28,29,33,36)(30,40,39,41)(35,37,38,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999729);